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Old page wikitext, before the edit (old_wikitext) | '{{Short description|Branch of mathematics}}
In [[mathematics]], '''homotopy theory''' is a systematic study of situations in which [[Map (mathematics)|maps]] can come with [[homotopy|homotopies]] between them. It originated as a topic in [[algebraic topology]], but nowadays is learned as an independent discipline.
== Applications to other fields of mathematics ==
Besides algebraic topology, the theory has also been used in other areas of mathematics such as:
* [[Algebraic geometry]] (e.g., [[A1 homotopy theory|A<sup>1</sup> homotopy theory]])
* [[Category theory]] (specifically the study of [[higher category theory|higher categories]])
== Concepts ==
=== Spaces and maps ===
In homotopy theory and algebraic topology, the word "space" denotes a [[topological space]]. In order to avoid [[Pathological (mathematics)|pathologies]], one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being [[Category of compactly generated weak Hausdorff spaces|compactly generated weak Hausdorff]] or a [[CW complex]].
In the same vein as above, a "[[Map (mathematics)|map]]" is a continuous function, possibly with some extra constraints.
Often, one works with a [[pointed space]]—that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserves basepoints; that is, it sends the basepoint of the domain to that of the codomain. In contrast, a free map is one which needn't preserve basepoints.
The Cartesian product of two pointed spaces <math>X, Y</math> are not naturally pointed. A substitute is the [[smash product]] <math>X \wedge Y</math> which is characterized by the [[adjoint functor|adjoint relation]]
:<math>\operatorname{Map}(X \wedge Y, Z) = \operatorname{Map}(X, \operatorname{Map}(Y, Z))</math>,
that is, a smash product is an analog of a [[tensor product]] in abstract algebra (see [[tensor-hom adjunction]]). Explicitly, <math>X \wedge Y</math> is the quotient of <math>X \times Y</math> by the [[wedge sum]] <math>X \vee Y</math>.
=== Homotopy ===
{{main|Homotopy}}
Let ''I'' denote the unit interval <math>[0, 1]</math>. A map
:<math>h: X \times I \to Y</math>
is called a [[homotopy]] from the map <math>h_0</math> to the map <math>h_1</math>, where <math>h_t(x) = h(x, t)</math>. Intuitively, we may think of <math>h</math> as a path from the map <math>h_0</math> to the map <math>h_1</math>. Indeed, a homotopy can be shown to be an [[equivalence relation]]. When ''X'', ''Y'' are pointed spaces, the maps <math>h_t</math> are required to preserve the basepoint and the homotopy <math>h</math> is called a [[based homotopy]]. A based homotopy is the same as a (based) map <math>X \wedge I_+ \to Y</math> where <math>I_+</math> is <math>I</math> together with a disjoint basepoint.<ref>{{harvnb|May|loc=Ch. 8. § 3.}}</ref>
Given a pointed space ''X'' and an [[integer]] <math>n \ge 0</math>, let <math>\pi_n X = [S^n, X]</math> be the homotopy classes of based maps <math>S^n \to X</math> from a (pointed) ''n''-sphere <math>S^n</math> to ''X''. As it turns out,
* for <math>n \ge 1</math>, <math>\pi_n X</math> are [[group (mathematics)|group]]s called [[homotopy group]]s; in particular, <math>\pi_1 X</math> is called the [[fundamental group]] of ''X'',
* for <math>n \ge 2</math>, <math>\pi_n X</math> are [[abelian group]]s by the [[Eckmann–Hilton argument]],
* <math>\pi_0 X</math> can be identified with the set of path-connected components in <math>X</math>.
Every group is the fundamental group of some space.<ref>{{harvnb|May|loc=Ch 4. § 5.}}</ref>
A map <math>f</math> is called a [[homotopy equivalence]] if there is another map <math>g</math> such that <math>f \circ g</math> and <math>g \circ f</math> are both homotopic to the identities. Two spaces are said to be homotopy equivalent if there is a homotopy equivalence between them. A homotopy equivalence class of spaces is then called a [[homotopy type]]. There is a weaker notion: a map <math>f : X \to Y</math> is said to be a [[weak homotopy equivalence]] if <math>f_* : \pi_n(X) \to \pi_n(Y)</math> is an isomorphism for each <math>n \ge 0</math> and each choice of a base point. A homotopy equivalence is a weak homotopy equivalence but the converse need not be true.
Through the adjunction
:<math>\operatorname{Map}(X \times I, Y) = \operatorname{Map}(X, \operatorname{Map}(I, Y)), \,\, h \mapsto (x \mapsto h(x, \cdot))</math>,
a homotopy <math>h : X \times I \to Y</math> is sometimes viewed as a map <math>X \to Y^I = \operatorname{Map}(I, Y)</math>.
=== CW complex ===
{{main|CW complex}}
A [[CW complex]] is a space that has a filtration <math>X \supset \cdots \supset X^n \supset X^{n-1} \supset \cdots \supset X^0</math> whose union is <math>X</math> and such that
# <math>X^0</math> is a discrete space, called the set of 0-cells (vertices) in <math>X</math>.
# Each <math>X^n</math> is obtained by attaching several ''n''-disks, ''n''-cells, to <math>X^{n-1}</math> via maps <math>S^{n-1} \to X^{n-1}</math>; i.e., the boundary of an n-disk is identified with the image of <math>S^{n-1}</math> in <math>X^{n-1}</math>.
# A subset <math>U</math> is open if and only if <math>U \cap X^n</math> is open for each <math>n</math>.
For example, a sphere <math>S^n</math> has two cells: one 0-cell and one <math>n</math>-cell, since <math>S^n</math> can be obtained by collapsing the boundary <math>S^{n-1}</math> of the ''n''-disk to a point. In general, every manifold has the homotopy type of a CW complex;<ref>{{harvnb|Milnor|1959|loc=Corollary 1}}. NB: "second countable" implies "separable".</ref> in fact, [[Morse theory]] implies that a compact manifold has the homotopy type of a finite CW complex.{{fact|date=September 2024}}
Remarkably, [[Whitehead's theorem]] says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing.
Another important result is the approximation theorem. First, the [[homotopy category]] of spaces is the category where an object is a space but a morphism is the homotopy class of a map. Then
{{math_theorem|name=[[CW approximation]]|math_statement=<ref>{{harvnb|May|loc=Ch. 10., § 5}}</ref> There exist a functor (called the CW approximation functor)
:<math>\Theta : \operatorname{Ho}(\textrm{spaces}) \to \operatorname{Ho}(\textrm{CW})</math>
from the homotopy category of spaces to the homotopy category of CW complexes as well as a natural transformation
:<math>\theta : i \circ \Theta \to \operatorname{Id},</math>
where <math>i : \operatorname{Ho}(\textrm{CW}) \hookrightarrow \operatorname{Ho}(\textrm{spaces})</math>, such that each <math>\theta_X : i(\Theta(X)) \to X</math> is a weak homotopy equivalence.
Similar statements also hold for pairs and excisive triads.<ref>{{harvnb|May|loc=Ch. 10., § 6}}</ref><ref>{{harvnb|May|loc=Ch. 10., § 7}}</ref>}}
Explicitly, the above approximation functor can be defined as the composition of the [[singular chain]] functor <math>S_*</math> followed by the geometric realization functor; see {{section link||Simplicial set}}.
The above theorem justifies a common habit of working only with CW complexes. For example, given a space <math>X</math>, one can just define the homology of <math>X</math> to the homology of the CW approximation of <math>X</math> (the cell structure of a CW complex determines the natural homology, the [[cellular homology]] and that can be taken to be the homology of the complex.)
=== Cofibration and fibration ===
A map <math>f: A \to X</math> is called a [[cofibration]] if given:
# A map <math>h_0 : X \to Z</math>, and
# A homotopy <math>g_t : A \to Z</math>
such that <math>h_0 \circ f = g_0</math>, there exists a homotopy <math>h_t : X \to Z</math> that extends <math>h_0</math> and such that <math>h_t \circ f = g_t</math>. An example is a [[neighborhood deformation retract]]; that is, <math>X</math> contains a [[mapping cylinder]] neighborhood of a closed subspace <math>A</math> and <math>f</math> the inclusion (e.g., a [[tubular neighborhood]] of a closed submanifold).<ref>{{harvnb|Hatcher|loc=Example 0.15.}}</ref> In fact, a cofibration can be characterized as a neighborhood deformation retract pair.<ref>{{harvnb|May|loc=Ch 6. § 4.}}</ref> Another basic example is a [[CW pair]] <math>(X, A)</math>; many often work only with CW complexes and the notion of a cofibration there is then often implicit.
A [[fibration]] in the sense of Hurewicz is the dual notion of a cofibration: that is, a map <math>p : X \to B</math> is a fibration if given (1) a map <math>h_0 : Z \to X</math> and (2) a homotopy <math>g_t : Z \to B</math> such that <math>p \circ h_0 = g_0</math>, there exists a homotopy <math>h_t: Z \to X</math> that extends <math>h_0</math> and such that <math>p \circ h_t = g_t</math>.
While a cofibration is characterized by the existence of a retract, a fibration is characterized by the existence of a section called the [[path lifting]] as follows. Let <math>p': Np \to B^I</math> be the pull-back of a map <math>p : E \to B</math> along <math>\chi \mapsto \chi(1) : B^I \to B</math>, called the [[mapping path space]] of <math>p</math>.<ref>Some authors use <math>\chi \mapsto \chi(0)</math>. The definition here is from {{harvnb|May|loc=Ch. 8., § 5.}}</ref> Viewing <math>p'</math> as a homotopy <math>N p\times I \to B</math>, if <math>p</math> is a fibration, then <math>p'</math> gives a homotopy <ref>{{harvnb|May|loc=Ch. 7., § 2.}}</ref>
:<math>s: Np \to E^I</math>
such that <math>s(e, \chi)(1) = e, \, (p^I \circ s)(e, \chi) = \chi</math> where <math>p^I : E^I \to B^I</math> is given by <math>p</math>.<ref><math>p</math> in the reference should be <math>p^I</math>.</ref> This <math>s</math> is called the path lifting associated to <math>p</math>. Conversely, if there is a path lifting <math>s</math>, then <math>p</math> is a fibration as a required homotopy is obtained via <math>s</math>.
A basic example of a fibration is a [[covering map]] since it comes a unique path lifting. If <math>E</math> is a [[principal bundle|principal ''G''-bundle]] over a paracompact space, that is, a space with a [[Group action#Remarkable properties of actions|free and transitive]] (topological) [[group action]] of a ([[topological group|topological]]) group, then the projection map <math>p: E \to X</math> is an example of a fibration, because a Hurewicz fibration can be checked locally on a paracompact space.<ref>{{harvnb|May|loc=Ch. 7., § 4.}}</ref>
There are also based versions of a cofibration and a fibration (namely, the maps are required to be based).<ref>{{harvnb|May|loc=Ch 8. § 3. and § 5.}}</ref>
=== Lifting property ===
A pair of maps <math>i : A \to X</math> and <math>p : E \to B</math> is said to satisfy the [[lifting property]] if for each commutative square diagram
:[[File:Lifting property diagram.png|frameless|150px]]
there is a map <math>\lambda</math> that makes the above diagram still commute. (The notion originates in the theory of [[model category|model categories]].)
Let <math>\mathfrak{c}</math> be a class of maps. Then a map <math>p : E \to B</math> is said to satisfy the [[right lifting property]] or the RLP if <math>p</math> satisfies the above lifting property for each <math>i</math> in <math>\mathfrak{c}</math>. Similarly, a map <math>i : A \to X</math> is said to satisfy the [[left lifting property]] or the LLP if it satisfies the lifting property for all the maps <math>p</math> in <math>\mathfrak{c}</math>.
For example, a Hurewicz fibration is exactly a map <math>p : E \to B</math> that satisfies the RLP for the inclusions <math>i_0 : A \to A \times I</math>. A [[Serre fibration]] is a map satisfying the RLP for the inclusions <math>i : S^{n - 1} \to D^n</math> where <math>S^{-1}</math> is the empty set. A Hurewicz fibration is a Serre fibration and the converse holds for CW complexes.<ref>https://ncatlab.org/nlab/show/a+Serre+fibration+between+CW-complexes+is+a+Hurewicz+fibration</ref>
On the other hand, a cofibration is a map satisfying the LLP for evaluation maps <math>p: B^I \to B</math> at <math>0</math>.
=== Loop and suspension ===
On the category of pointed spaces, there are two important functors: the [[loop functor]] <math>\Omega</math> and the (reduced) [[suspension functor]] <math>\Sigma</math>, which are in the [[adjoint functor|adjoint relation]]. Precisely, they are defined as<ref>{{harvnb|May|loc=Ch. 8, § 2.}}</ref>
*<math>\Omega X = \operatorname{Map}(S^1, X)</math>, and
*<math>\Sigma X = X \wedge S^1</math>.
Because of the adjoint relation between a smash product and a mapping space, we have:
:<math>\operatorname{Map}(\Sigma X, Y) = \operatorname{Map}(X, \Omega Y).</math>
These functors are used to construct [[fiber sequence]]s and [[cofiber sequence]]s. Namely, if <math>f : X \to Y</math> is a map, the fiber sequence generated by <math>f</math> is the exact sequence<ref>{{harvnb|May|loc=Ch. 8, § 6.}}</ref>
:<math>\cdots \to \Omega^2 Ff \to \Omega^2 X \to \Omega^2 Y \to \Omega Ff \to \Omega X \to \Omega Y \to Ff \to X \to Y</math>
where <math>Ff</math> is the [[homotopy fiber]] of <math>f</math>; i.e., a fiber obtained after replacing <math>f</math> by a (based) fibration. The cofibration sequence generated by <math>f</math> is <math>X \to Y \to C f \to \Sigma X \to \cdots,</math> where <math>Cf</math> is the homotooy cofiber of <math>f</math> constructed like a homotopy fiber (use a quotient instead of a fiber.)
The functors <math>\Omega, \Sigma</math> restrict to the category of CW complexes in the following weak sense: a theorem of Milnor says that if <math>X</math> has the homotopy type of a CW complex, then so does its loop space <math>\Omega X</math>.<ref>{{harvnb|Milnor|1959|loc=Theorem 3.}}</ref>
=== Classifying spaces and homotopy operations ===
Given a topological group ''G'', the [[classifying space]] for [[principal bundle|principal ''G''-bundles]] ("the" up to equivalence) is a space <math>BG</math> such that, for each space ''X'',
:<math>[X, BG] = </math> {principal ''G''-bundle on ''X''} / ~ <math>, \,\, [f] \mapsto [f^* EG]</math>
where
*the left-hand side is the set of homotopy classes of maps <math>X \to BG</math>,
*~ refers isomorphism of bundles, and
*= is given by pulling-back the distinguished bundle <math>EG</math> on <math>BG</math> (called universal bundle) along a map <math>X \to BG</math>.
[[Brown's representability theorem]] guarantees the existence of classifying spaces.
=== Spectrum and generalized cohomology ===
{{main|Spectrum (algebraic topology)|Generalized cohomology}}
The idea that a classifying space classifies principal bundles can be pushed further. For example, one might try to classify cohomology classes: given an [[abelian group]] ''A'' (such as <math>\mathbb{Z}</math>),
:<math>[X, K(A, n)] = \operatorname{H}^n(X; A)</math>
where <math>K(A, n)</math> is the [[Eilenberg–MacLane space]]. The above equation leads to the notion of a generalized cohomology theory; i.e., a [[contravariant functor]] from the category of spaces to the [[category of abelian groups]] that satisfies the axioms generalizing ordinary cohomology theory. As it turns out, such a functor may not be [[representable functor|representable]] by a space but it can always be represented by a sequence of (pointed) spaces with structure maps called a spectrum. In other words, to give a generalized cohomology theory is to give a spectrum. A [[K-theory]] is an example of a generalized cohomology theory.
A basic example of a spectrum is a [[sphere spectrum]]: <math>S^0 \to S^1 \to S^2 \to \cdots</math>
== Key theorems ==
*[[Seifert–van Kampen theorem]]
*[[Homotopy excision theorem]]
*[[Freudenthal suspension theorem]] (a corollary of the excision theorem)
*[[Landweber exact functor theorem]]
*[[Dold–Kan correspondence]]
*[[Eckmann–Hilton argument]] - this shows for instance higher homotopy groups are [[abelian group|abelian]].
*[[Universal coefficient theorem]]
*[[Dold–Thom theorem]]
== Obstruction theory and characteristic class ==
{{expand section|date=May 2020}}
See also: [[Characteristic class]], [[Postnikov tower]], [[Whitehead torsion]]
== Localization and completion of a space ==
{{expand section|date=May 2020}}
{{main|Localization of a topological space}}
== Specific theories ==
There are several specific theories
*[[simple homotopy theory]]
*[[stable homotopy theory]]
*[[chromatic homotopy theory]]
*[[rational homotopy theory]]
*[[p-adic homotopy theory]]
*[[equivariant homotopy theory]]
== Homotopy hypothesis ==
{{main|Homotopy hypothesis}}
One of the basic questions in the foundations of homotopy theory is the nature of a space. The [[homotopy hypothesis]] asks whether a space is something fundamentally algebraic.
If one prefers to work with a space instead of a pointed space, there is the notion of a [[fundamental groupoid]] (and higher variants): by definition, the fundamental groupoid of a space ''X'' is the [[category (mathematics)|category]] where the [[object (category theory)|objects]] are the points of ''X'' and the [[morphism]]s are paths.
== Abstract homotopy theory ==
Abstract homotopy theory is an axiomatic approach to homotopy theory. Such axiomatization is useful for non-traditional applications of homotopy theory. One approach to axiomatization is by Quillen's [[model category|model categories]]. A model category is a category with a choice of three classes of maps called weak equivalences, cofibrations and fibrations, subject to the axioms that are reminiscent of facts in algebraic topology. For example, the category of (reasonable) topological spaces has a structure of a model category where a weak equivalence is a weak homotopy equivalence, a cofibration a certain retract and a fibration a Serre fibration.<ref>{{harvnb|Dwyer|Spalinski|loc=Example 3.5.}}</ref> Another example is the category of non-negatively graded chain complexes over a fixed base ring.<ref>{{harvnb|Dwyer|Spalinski|loc=Example 3.7.}}</ref>
See also: [[Algebraic homotopy]]
=== Simplicial set ===
{{main|Simplicial set|simplicial homotopy theory}}
A [[simplicial set]] is an abstract generalization of a [[simplicial complex]] and can play a role of a "space" in some sense. Despite the name, it is not a set but is a sequence of sets together with the certain maps (face and degeneracy) between those sets.
For example, given a space <math>X</math>, for each integer <math>n \ge 0</math>, let <math>S_n X</math> be the set of all maps from the ''n''-simplex to <math>X</math>. Then the sequence <math>S_n X</math> of sets is a simplicial set.<ref name="May simplicial">{{harvnb|May|loc=Ch. 16, § 4.}}</ref> Each simplicial set <math>K = \{ K_n \}_{n \ge 0}</math> has a naturally associated chain complex and the homology of that chain complex is the homology of <math>K</math>. The [[singular homology]] of <math>X</math> is precisely the homology of the simplicial set <math>S_* X</math>. Also, the [[Simplicial_set#Geometric_realization|geometric realization]] <math>| \cdot |</math> of a simplicial set is a CW complex and the composition <math>X \mapsto |S_* X|</math> is precisely the CW approximation functor.
Another important example is a category or more precisely the [[nerve of a category]], which is a simplicial set. In fact, a simplicial set is the nerve of some category if and only if it satisfies the [[Segal condition]]s (a theorem of Grothendieck). Each category is completely determined by its nerve. In this way, a category can be viewed as a special kind of a simplicial set, and this observation is used to generalize a category. Namely, an [[infinity category|<math>\infty</math>-category]] or an [[infinity groupoid|<math>\infty</math>-groupoid]] is defined as particular kinds of simplicial sets.
Since simplicial sets are sort of abstract spaces (if not topological spaces), it is possible to develop the homotopy theory on them, which is called the [[simplicial homotopy theory]].<ref name="May simplicial" />
== See also ==
*[[Highly structured ring spectrum]]
*[[Homotopy type theory]]
*[[Pursuing Stacks]]
*[[Shape theory (mathematics)|Shape theory]]
*[[Moduli stack of formal group laws]]
== References ==
{{reflist}}
*May, J. [http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf A Concise Course in Algebraic Topology]
* J. P. May and K. Ponto, More Concise Algebraic Topology: Localization, completion, and model categories
*{{cite book|author=George William Whitehead|author-link=George W. Whitehead|title=Elements of homotopy theory|url=https://books.google.com/books?id=wlrvAAAAMAAJ|access-date=September 6, 2011|edition=3rd|series=Graduate Texts in Mathematics|volume=61|year=1978|publisher=Springer-Verlag|location=New York-Berlin|isbn=978-0-387-90336-1|pages=xxi+744|mr=0516508 }}
*Ronald Brown, ''[http://arquivo.pt/wayback/20160514115224/http://www.bangor.ac.uk/r.brown/topgpds.html Topology and groupoids]'' (2006) Booksurge LLC {{ISBN|1-4196-2722-8}}.
* https://ncatlab.org/nlab/show/homotopical+algebra
* Homotopy Theories and Model Categories by W.G. Dwyer and J. Spalinski in [https://books.google.com/books?id=xoM5DxQZihQC&printsec=copyright#v=onepage&q&f=false Handbook of Algebraic Topology] edited by I.M. James
*{{cite web |first=Allen |last=Hatcher |url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html |title=Algebraic topology}}
*{{cite journal |last1=Milnor |first1=John |title=On spaces having the homotopy type of 𝐶𝑊-complex |journal=Transactions of the American Mathematical Society |date=1959 |volume=90 |issue=2 |pages=272–280 |doi=10.1090/S0002-9947-1959-0100267-4 |url=https://www.semanticscholar.org/paper/On-spaces-having-the-homotopy-type-of-a-CW-complex-Milnor/905bb7242d4e2b7b7e168d12718b6595c98e98d9 |language=en |issn=0002-9947}}
* Edwin Spanier, Algebraic topology
* Dennis Sullivan. Genetics of homotopy theory and the Adams conjecture. Ann. of Math. (2), 100:1–79, 1974.
== Further reading ==
*[http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf Cisinski's notes]
*http://ncatlab.org/nlab/files/Abstract-Homotopy.pdf
*[https://uregina.ca/~franklam/Math527/Math527.html Math 527 - Homotopy Theory Spring 2013, Section F1], lectures by Martin Frankland
* D. Quillen, Homotopical algebra, Lectures Notes in Math. vol. 43, Springer Verlag, 1967.
* https://ncatlab.org/nlab/show/homotopy+theory
== External links ==
{{cite web|title=homotopy theory |url=https://ncatlab.org/nlab/show/homotopy+theory |website=ncatlab.org}}
{{Authority control}}
[[Category:Homotopy theory| ]]' |
New page wikitext, after the edit (new_wikitext) | '{{Short description|Branch of mathematics}}
In [[mathematics]], '''homotopy theory''' is a systematic study of situations in which [[Map (mathematics)|maps]] can come with [[homotopy|homotopies]] between them. It originated as a topic in [[algebraic topology]], but nowadays is learned as an independent discipline.
== Applications to other fields of mathematics ==
Besides algebraic topology, the theory has also been used in other areas of mathematics such as:
* [[Algebraic geometry]] (e.g., [[A1 homotopy theory|A<sup>1</sup> homotopy theory]])
* [[Category theory]] (specifically the study of [[higher category theory|higher categories]])
== Concepts ==
=== Spaces and maps ===
In homotopy theory and algebraic topology, the word "space" denotes a [[topological space]]. In order to avoid [[Pathological (mathematics)|pathologies]], one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being [[Category of compactly generated weak Hausdorff spaces|compactly generated weak Hausdorff]] or a [[CW complex]].
In the same vein as above, a "[[Map (mathematics)|map]]" is a continuous function, possibly with some extra constraints.
Often, one works with a [[pointed space]]—that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserves basepoints; that is, it sends the basepoint of the domain to that of the codomain. In contrast, a free map is one which needn't preserve basepoints.
The Cartesian product of two pointed spaces <math>X, Y</math> are not naturally pointed. A substitute is the [[smash product]] <math>X \wedge Y</math> which is characterized by the [[adjoint functor|adjoint relation]]
:<math>\operatorname{Map}(X \wedge Y, Z) = \operatorname{Map}(X, \operatorname{Map}(Y, Z))</math>,
that is, a smash product is an analog of a [[tensor product]] in abstract algebra (see [[tensor-hom adjunction]]). Explicitly, <math>X \wedge Y</math> is the quotient of <math>X \times Y</math> by the [[wedge sum]] <math>X \vee Y</math>.
=== Homotopy ===
{{main|Homotopy}}
Let ''I'' denote the unit interval <math>[0, 1]</math>. A map
:<math>h: X \times I \to Y</math>
is called a [[homotopy]] from the map <math>h_0</math> to the map <math>h_1</math>, where <math>h_t(x) = h(x, t)</math>. Intuitively, we may think of <math>h</math> as a path from the map <math>h_0</math> to the map <math>h_1</math>. Indeed, a homotopy can be shown to be an [[equivalence relation]]. When ''X'', ''Y'' are pointed spaces, the maps <math>h_t</math> are required to preserve the basepoint and the homotopy <math>h</math> is called a [[based homotopy]]. A based homotopy is the same as a (based) map <math>X \wedge I_+ \to Y</math> where <math>I_+</math> is <math>I</math> together with a disjoint basepoint.<ref>{{harvnb|May|loc=Ch. 8. § 3.}}</ref>
Given a pointed space ''X'' and an [[integer]] <math>n \ge 0</math>, let <math>\pi_n X = [S^n, X]</math> be the homotopy classes of based maps <math>S^n \to X</math> from a (pointed) ''n''-sphere <math>S^n</math> to ''X''. As it turns out,
* for <math>n \ge 1</math>, <math>\pi_n X</math> are [[group (mathematics)|group]]s called [[homotopy group]]s; in particular, <math>\pi_1 X</math> is called the [[fundamental group]] of ''X'',
* for <math>n \ge 2</math>, <math>\pi_n X</math> are [[abelian group]]s by the [[Eckmann–Hilton argument]],
* <math>\pi_0 X</math> can be identified with the set of path-connected components in <math>X</math>.
Every group is the fundamental group of some space.<ref>{{harvnb|May|loc=Ch 4. § 5.}}</ref>
A map <math>f</math> is called a [[homotopy equivalence]] if there is another map <math>g</math> such that <math>f \circ g</math> and <math>g \circ f</math> are both homotopic to the identities. Two spaces are said to be homotopy equivalent if there is a homotopy equivalence between them. A homotopy equivalence class of spaces is then called a [[homotopy type]]. There is a weaker notion: a map <math>f : X \to Y</math> is said to be a [[weak homotopy equivalence]] if <math>f_* : \pi_n(X) \to \pi_n(Y)</math> is an isomorphism for each <math>n \ge 0</math> and each choice of a base point. A homotopy equivalence is a weak homotopy equivalence but the converse need not be true.
Through the adjunction
:<math>\operatorname{Map}(X \times I, Y) = \operatorname{Map}(X, \operatorname{Map}(I, Y)), \,\, h \mapsto (x \mapsto h(x, \cdot))</math>,
a homotopy <math>h : X \times I \to Y</math> is sometimes viewed as a map <math>X \to Y^I = \operatorname{Map}(I, Y)</math>.
=== CW complex ===
{{main|CW complex}}
A [[CW complex]] is a space that has a filtration <math>X \supset \cdots \supset X^n \supset X^{n-1} \supset \cdots \supset X^0</math> whose union is <math>X</math> and such that
# <math>X^0</math> is a discrete space, called the set of 0-cells (vertices) in <math>X</math>.
# Each <math>X^n</math> is obtained by attaching several ''n''-disks, ''n''-cells, to <math>X^{n-1}</math> via maps <math>S^{n-1} \to X^{n-1}</math>; i.e., the boundary of an n-disk is identified with the image of <math>S^{n-1}</math> in <math>X^{n-1}</math>.
# A subset <math>U</math> is open if and only if <math>U \cap X^n</math> is open for each <math>n</math>.
For example, a sphere <math>S^n</math> has two cells: one 0-cell and one <math>n</math>-cell, since <math>S^n</math> can be obtained by collapsing the boundary <math>S^{n-1}</math> of the ''n''-disk to a point. In general, every manifold has the homotopy type of a CW complex;<ref>{{harvnb|Milnor|1959|loc=Corollary 1}}. NB: "second countable" implies "separable".</ref> in fact, [[Morse theory]] implies that a compact manifold has the homotopy type of a finite CW complex.{{fact|date=September 2024}}
Remarkably, [[Whitehead's theorem]] says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing.
Another important result is the approximation theorem. First, the [[homotopy category]] of spaces is the category where an object is a space but a morphism is the homotopy class of a map. Then
{{math_theorem|name=[[CW approximation]]|math_statement=<ref>{{harvnb|May|loc=Ch. 10., § 5}}</ref> There exist a functor (called the CW approximation functor)
:<math>\Theta : \operatorname{Ho}(\textrm{spaces}) \to \operatorname{Ho}(\textrm{CW})</math>
from the homotopy category of spaces to the homotopy category of CW complexes as well as a natural transformation
:<math>\theta : i \circ \Theta \to \operatorname{Id},</math>
where <math>i : \operatorname{Ho}(\textrm{CW}) \hookrightarrow \operatorname{Ho}(\textrm{spaces})</math>, such that each <math>\theta_X : i(\Theta(X)) \to X</math> is a weak homotopy equivalence.
Similar statements also hold for pairs and excisive triads.<ref>{{harvnb|May|loc=Ch. 10., § 6}}</ref><ref>{{harvnb|May|loc=Ch. 10., § 7}}</ref>}}
Explicitly, the above approximation functor can be defined as the composition of the [[singular chain]] functor <math>S_*</math> followed by the geometric realization functor; see {{section link||Simplicial set}}.
The above theorem justifies a common habit of working only with CW complexes. For example, given a space <math>X</math>, one can just define the homology of <math>X</math> to the homology of the CW approximation of <math>X</math> (the cell structure of a CW complex determines the natural homology, the [[cellular homology]] and that can be taken to be the homology of the complex.)
=== Cofibration and fibration ===
A map <math>f: A \to X</math> is called a [[cofibration]] if given:
# A map <math>h_0 : X \to Z</math>, and
# A homotopy <math>g_t : A \to Z</math>
such that <math>h_0 \circ f = g_0</math>, there exists a homotopy <math>h_t : X \to Z</math> that extends <math>h_0</math> and such that <math>h_t \circ f = g_t</math>. An example is a [[neighborhood deformation retract]]; that is, <math>X</math> contains a [[mapping cylinder]] neighborhood of a closed subspace <math>A</math> and <math>f</math> the inclusion (e.g., a [[tubular neighborhood]] of a closed submanifold).<ref>{{harvnb|Hatcher|loc=Example 0.15.}}</ref> In fact, a cofibration can be characterized as a neighborhood deformation retract pair.<ref>{{harvnb|May|loc=Ch 6. § 4.}}</ref> Another basic example is a [[CW pair]] <math>(X, A)</math>; many often work only with CW complexes and the notion of a cofibration there is then often implicit.
A [[fibration]] in the sense of Hurewicz is the dual notion of a cofibration: that is, a map <math>p : X \to B</math> is a fibration if given (1) a map <math>h_0 : Z \to X</math> and (2) a homotopy <math>g_t : Z \to B</math> such that <math>p \circ h_0 = g_0</math>, there exists a homotopy <math>h_t: Z \to X</math> that extends <math>h_0</math> and such that <math>p \circ h_t = g_t</math>.
While a cofibration is characterized by the existence of a retract, a fibration is characterized by the existence of a section called the [[path lifting]] as follows. Let <math>p': Np \to B^I</math> be the pull-back of a map <math>p : E \to B</math> along <math>\chi \mapsto \chi(1) : B^I \to B</math>, called the [[mapping path space]] of <math>p</math>.<ref>Some authors use <math>\chi \mapsto \chi(0)</math>. The definition here is from {{harvnb|May|loc=Ch. 8., § 5.}}</ref> Viewing <math>p'</math> as a homotopy <math>N p\times I \to B</math>, if <math>p</math> is a fibration, then <math>p'</math> gives a homotopy <ref>{{harvnb|May|loc=Ch. 7., § 2.}}</ref>
:<math>s: Np \to E^I</math>
such that <math>s(e, \chi)(1) = e, \, (p^I \circ s)(e, \chi) = \chi</math> where <math>p^I : E^I \to B^I</math> is given by <math>p</math>.<ref><math>p</math> in the reference should be <math>p^I</math>.</ref> This <math>s</math> is called the path lifting associated to <math>p</math>. Conversely, if there is a path lifting <math>s</math>, then <math>p</math> is a fibration as a required homotopy is obtained via <math>s</math>.
A basic example of a fibration is a [[covering map]] since it comes a unique path lifting. If <math>E</math> is a [[principal bundle|principal ''G''-bundle]] over a paracompact space, that is, a space with a [[Group action#Remarkable properties of actions|free and transitive]] (topological) [[group action]] of a ([[topological group|topological]]) group, then the projection map <math>p: E \to X</math> is an example of a fibration, because a Hurewicz fibration can be checked locally on a paracompact space.<ref>{{harvnb|May|loc=Ch. 7., § 4.}}</ref>
There are also based versions of a cofibration and a fibration (namely, the maps are required to be based).<ref>{{harvnb|May|loc=Ch 8. § 3. and § 5.}}</ref>
=== Lifting property ===
A pair of maps <math>i : A \to X</math> and <math>p : E \to B</math> is said to satisfy the [[lifting property]] <ref>{{harvnb|May|Ponto|loc=Definition 14.1.5.}}</ref> if for each commutative square diagram
:[[File:Lifting property diagram.png|frameless|150px]]
there is a map <math>\lambda</math> that makes the above diagram still commute. (The notion originates in the theory of [[model category|model categories]].)
Let <math>\mathfrak{c}</math> be a class of maps. Then a map <math>p : E \to B</math> is said to satisfy the [[right lifting property]] or the RLP if <math>p</math> satisfies the above lifting property for each <math>i</math> in <math>\mathfrak{c}</math>. Similarly, a map <math>i : A \to X</math> is said to satisfy the [[left lifting property]] or the LLP if it satisfies the lifting property for all the maps <math>p</math> in <math>\mathfrak{c}</math>.
For example, a Hurewicz fibration is exactly a map <math>p : E \to B</math> that satisfies the RLP for the inclusions <math>i_0 : A \to A \times I</math>. A [[Serre fibration]] is a map satisfying the RLP for the inclusions <math>i : S^{n - 1} \to D^n</math> where <math>S^{-1}</math> is the empty set. A Hurewicz fibration is a Serre fibration and the converse holds for CW complexes.<ref>https://ncatlab.org/nlab/show/a+Serre+fibration+between+CW-complexes+is+a+Hurewicz+fibration</ref>
On the other hand, a cofibration is a map satisfying the LLP for evaluation maps <math>p: B^I \to B</math> at <math>0</math>.
=== Loop and suspension ===
On the category of pointed spaces, there are two important functors: the [[loop functor]] <math>\Omega</math> and the (reduced) [[suspension functor]] <math>\Sigma</math>, which are in the [[adjoint functor|adjoint relation]]. Precisely, they are defined as<ref>{{harvnb|May|loc=Ch. 8, § 2.}}</ref>
*<math>\Omega X = \operatorname{Map}(S^1, X)</math>, and
*<math>\Sigma X = X \wedge S^1</math>.
Because of the adjoint relation between a smash product and a mapping space, we have:
:<math>\operatorname{Map}(\Sigma X, Y) = \operatorname{Map}(X, \Omega Y).</math>
These functors are used to construct [[fiber sequence]]s and [[cofiber sequence]]s. Namely, if <math>f : X \to Y</math> is a map, the fiber sequence generated by <math>f</math> is the exact sequence<ref>{{harvnb|May|loc=Ch. 8, § 6.}}</ref>
:<math>\cdots \to \Omega^2 Ff \to \Omega^2 X \to \Omega^2 Y \to \Omega Ff \to \Omega X \to \Omega Y \to Ff \to X \to Y</math>
where <math>Ff</math> is the [[homotopy fiber]] of <math>f</math>; i.e., a fiber obtained after replacing <math>f</math> by a (based) fibration. The cofibration sequence generated by <math>f</math> is <math>X \to Y \to C f \to \Sigma X \to \cdots,</math> where <math>Cf</math> is the homotooy cofiber of <math>f</math> constructed like a homotopy fiber (use a quotient instead of a fiber.)
The functors <math>\Omega, \Sigma</math> restrict to the category of CW complexes in the following weak sense: a theorem of Milnor says that if <math>X</math> has the homotopy type of a CW complex, then so does its loop space <math>\Omega X</math>.<ref>{{harvnb|Milnor|1959|loc=Theorem 3.}}</ref>
=== Classifying spaces and homotopy operations ===
Given a topological group ''G'', the [[classifying space]] for [[principal bundle|principal ''G''-bundles]] ("the" up to equivalence) is a space <math>BG</math> such that, for each space ''X'',
:<math>[X, BG] = </math> {principal ''G''-bundle on ''X''} / ~ <math>, \,\, [f] \mapsto [f^* EG]</math>
where
*the left-hand side is the set of homotopy classes of maps <math>X \to BG</math>,
*~ refers isomorphism of bundles, and
*= is given by pulling-back the distinguished bundle <math>EG</math> on <math>BG</math> (called universal bundle) along a map <math>X \to BG</math>.
[[Brown's representability theorem]] guarantees the existence of classifying spaces.
=== Spectrum and generalized cohomology ===
{{main|Spectrum (algebraic topology)|Generalized cohomology}}
The idea that a classifying space classifies principal bundles can be pushed further. For example, one might try to classify cohomology classes: given an [[abelian group]] ''A'' (such as <math>\mathbb{Z}</math>),
:<math>[X, K(A, n)] = \operatorname{H}^n(X; A)</math>
where <math>K(A, n)</math> is the [[Eilenberg–MacLane space]]. The above equation leads to the notion of a generalized cohomology theory; i.e., a [[contravariant functor]] from the category of spaces to the [[category of abelian groups]] that satisfies the axioms generalizing ordinary cohomology theory. As it turns out, such a functor may not be [[representable functor|representable]] by a space but it can always be represented by a sequence of (pointed) spaces with structure maps called a spectrum. In other words, to give a generalized cohomology theory is to give a spectrum. A [[K-theory]] is an example of a generalized cohomology theory.
A basic example of a spectrum is a [[sphere spectrum]]: <math>S^0 \to S^1 \to S^2 \to \cdots</math>
== Key theorems ==
*[[Seifert–van Kampen theorem]]
*[[Homotopy excision theorem]]
*[[Freudenthal suspension theorem]] (a corollary of the excision theorem)
*[[Landweber exact functor theorem]]
*[[Dold–Kan correspondence]]
*[[Eckmann–Hilton argument]] - this shows for instance higher homotopy groups are [[abelian group|abelian]].
*[[Universal coefficient theorem]]
*[[Dold–Thom theorem]]
== Obstruction theory and characteristic class ==
{{expand section|date=May 2020}}
See also: [[Characteristic class]], [[Postnikov tower]], [[Whitehead torsion]]
== Localization and completion of a space ==
{{expand section|date=May 2020}}
{{main|Localization of a topological space}}
== Specific theories ==
There are several specific theories
*[[simple homotopy theory]]
*[[stable homotopy theory]]
*[[chromatic homotopy theory]]
*[[rational homotopy theory]]
*[[p-adic homotopy theory]]
*[[equivariant homotopy theory]]
== Homotopy hypothesis ==
{{main|Homotopy hypothesis}}
One of the basic questions in the foundations of homotopy theory is the nature of a space. The [[homotopy hypothesis]] asks whether a space is something fundamentally algebraic.
If one prefers to work with a space instead of a pointed space, there is the notion of a [[fundamental groupoid]] (and higher variants): by definition, the fundamental groupoid of a space ''X'' is the [[category (mathematics)|category]] where the [[object (category theory)|objects]] are the points of ''X'' and the [[morphism]]s are paths.
== Abstract homotopy theory ==
Abstract homotopy theory is an axiomatic approach to homotopy theory. Such axiomatization is useful for non-traditional applications of homotopy theory. One approach to axiomatization is by Quillen's [[model category|model categories]]. A model category is a category with a choice of three classes of maps called weak equivalences, cofibrations and fibrations, subject to the axioms that are reminiscent of facts in algebraic topology. For example, the category of (reasonable) topological spaces has a structure of a model category where a weak equivalence is a weak homotopy equivalence, a cofibration a certain retract and a fibration a Serre fibration.<ref>{{harvnb|Dwyer|Spalinski|loc=Example 3.5.}}</ref> Another example is the category of non-negatively graded chain complexes over a fixed base ring.<ref>{{harvnb|Dwyer|Spalinski|loc=Example 3.7.}}</ref>
See also: [[Algebraic homotopy]]
=== Simplicial set ===
{{main|Simplicial set|simplicial homotopy theory}}
A [[simplicial set]] is an abstract generalization of a [[simplicial complex]] and can play a role of a "space" in some sense. Despite the name, it is not a set but is a sequence of sets together with the certain maps (face and degeneracy) between those sets.
For example, given a space <math>X</math>, for each integer <math>n \ge 0</math>, let <math>S_n X</math> be the set of all maps from the ''n''-simplex to <math>X</math>. Then the sequence <math>S_n X</math> of sets is a simplicial set.<ref name="May simplicial">{{harvnb|May|loc=Ch. 16, § 4.}}</ref> Each simplicial set <math>K = \{ K_n \}_{n \ge 0}</math> has a naturally associated chain complex and the homology of that chain complex is the homology of <math>K</math>. The [[singular homology]] of <math>X</math> is precisely the homology of the simplicial set <math>S_* X</math>. Also, the [[Simplicial_set#Geometric_realization|geometric realization]] <math>| \cdot |</math> of a simplicial set is a CW complex and the composition <math>X \mapsto |S_* X|</math> is precisely the CW approximation functor.
Another important example is a category or more precisely the [[nerve of a category]], which is a simplicial set. In fact, a simplicial set is the nerve of some category if and only if it satisfies the [[Segal condition]]s (a theorem of Grothendieck). Each category is completely determined by its nerve. In this way, a category can be viewed as a special kind of a simplicial set, and this observation is used to generalize a category. Namely, an [[infinity category|<math>\infty</math>-category]] or an [[infinity groupoid|<math>\infty</math>-groupoid]] is defined as particular kinds of simplicial sets.
Since simplicial sets are sort of abstract spaces (if not topological spaces), it is possible to develop the homotopy theory on them, which is called the [[simplicial homotopy theory]].<ref name="May simplicial" />
== See also ==
*[[Highly structured ring spectrum]]
*[[Homotopy type theory]]
*[[Pursuing Stacks]]
*[[Shape theory (mathematics)|Shape theory]]
*[[Moduli stack of formal group laws]]
== References ==
{{reflist}}
*May, J. [http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf A Concise Course in Algebraic Topology]
* J. P. May and K. Ponto, More Concise Algebraic Topology: Localization, completion, and model categories
*{{cite book|author=George William Whitehead|author-link=George W. Whitehead|title=Elements of homotopy theory|url=https://books.google.com/books?id=wlrvAAAAMAAJ|access-date=September 6, 2011|edition=3rd|series=Graduate Texts in Mathematics|volume=61|year=1978|publisher=Springer-Verlag|location=New York-Berlin|isbn=978-0-387-90336-1|pages=xxi+744|mr=0516508 }}
*Ronald Brown, ''[http://arquivo.pt/wayback/20160514115224/http://www.bangor.ac.uk/r.brown/topgpds.html Topology and groupoids]'' (2006) Booksurge LLC {{ISBN|1-4196-2722-8}}.
* https://ncatlab.org/nlab/show/homotopical+algebra
* Homotopy Theories and Model Categories by W.G. Dwyer and J. Spalinski in [https://books.google.com/books?id=xoM5DxQZihQC&printsec=copyright#v=onepage&q&f=false Handbook of Algebraic Topology] edited by I.M. James
*{{cite web |first=Allen |last=Hatcher |url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html |title=Algebraic topology}}
*{{cite journal |last1=Milnor |first1=John |title=On spaces having the homotopy type of 𝐶𝑊-complex |journal=Transactions of the American Mathematical Society |date=1959 |volume=90 |issue=2 |pages=272–280 |doi=10.1090/S0002-9947-1959-0100267-4 |url=https://www.semanticscholar.org/paper/On-spaces-having-the-homotopy-type-of-a-CW-complex-Milnor/905bb7242d4e2b7b7e168d12718b6595c98e98d9 |language=en |issn=0002-9947}}
* Edwin Spanier, Algebraic topology
* Dennis Sullivan. Genetics of homotopy theory and the Adams conjecture. Ann. of Math. (2), 100:1–79, 1974.
== Further reading ==
*[http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf Cisinski's notes]
*http://ncatlab.org/nlab/files/Abstract-Homotopy.pdf
*[https://uregina.ca/~franklam/Math527/Math527.html Math 527 - Homotopy Theory Spring 2013, Section F1], lectures by Martin Frankland
* D. Quillen, Homotopical algebra, Lectures Notes in Math. vol. 43, Springer Verlag, 1967.
* https://ncatlab.org/nlab/show/homotopy+theory
== External links ==
{{cite web|title=homotopy theory |url=https://ncatlab.org/nlab/show/homotopy+theory |website=ncatlab.org}}
{{Authority control}}
[[Category:Homotopy theory| ]]' |
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=== Lifting property ===
-A pair of maps <math>i : A \to X</math> and <math>p : E \to B</math> is said to satisfy the [[lifting property]] if for each commutative square diagram
+A pair of maps <math>i : A \to X</math> and <math>p : E \to B</math> is said to satisfy the [[lifting property]] <ref>{{harvnb|May|Ponto|loc=Definition 14.1.5.}}</ref> if for each commutative square diagram
:[[File:Lifting property diagram.png|frameless|150px]]
there is a map <math>\lambda</math> that makes the above diagram still commute. (The notion originates in the theory of [[model category|model categories]].)
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Parsed HTML source of the new revision (new_html) | '<div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Branch of mathematics</div>
<p>In <a href="/https/en.wikipedia.org/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>homotopy theory</b> is a systematic study of situations in which <a href="/https/en.wikipedia.org/wiki/Map_(mathematics)" title="Map (mathematics)">maps</a> can come with <a href="/https/en.wikipedia.org/wiki/Homotopy" title="Homotopy">homotopies</a> between them. It originated as a topic in <a href="/https/en.wikipedia.org/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>, but nowadays is learned as an independent discipline.
</p>
<div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Applications_to_other_fields_of_mathematics"><span class="tocnumber">1</span> <span class="toctext">Applications to other fields of mathematics</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Concepts"><span class="tocnumber">2</span> <span class="toctext">Concepts</span></a>
<ul>
<li class="toclevel-2 tocsection-3"><a href="#Spaces_and_maps"><span class="tocnumber">2.1</span> <span class="toctext">Spaces and maps</span></a></li>
<li class="toclevel-2 tocsection-4"><a href="#Homotopy"><span class="tocnumber">2.2</span> <span class="toctext">Homotopy</span></a></li>
<li class="toclevel-2 tocsection-5"><a href="#CW_complex"><span class="tocnumber">2.3</span> <span class="toctext">CW complex</span></a></li>
<li class="toclevel-2 tocsection-6"><a href="#Cofibration_and_fibration"><span class="tocnumber">2.4</span> <span class="toctext">Cofibration and fibration</span></a></li>
<li class="toclevel-2 tocsection-7"><a href="#Lifting_property"><span class="tocnumber">2.5</span> <span class="toctext">Lifting property</span></a></li>
<li class="toclevel-2 tocsection-8"><a href="#Loop_and_suspension"><span class="tocnumber">2.6</span> <span class="toctext">Loop and suspension</span></a></li>
<li class="toclevel-2 tocsection-9"><a href="#Classifying_spaces_and_homotopy_operations"><span class="tocnumber">2.7</span> <span class="toctext">Classifying spaces and homotopy operations</span></a></li>
<li class="toclevel-2 tocsection-10"><a href="#Spectrum_and_generalized_cohomology"><span class="tocnumber">2.8</span> <span class="toctext">Spectrum and generalized cohomology</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-11"><a href="#Key_theorems"><span class="tocnumber">3</span> <span class="toctext">Key theorems</span></a></li>
<li class="toclevel-1 tocsection-12"><a href="#Obstruction_theory_and_characteristic_class"><span class="tocnumber">4</span> <span class="toctext">Obstruction theory and characteristic class</span></a></li>
<li class="toclevel-1 tocsection-13"><a href="#Localization_and_completion_of_a_space"><span class="tocnumber">5</span> <span class="toctext">Localization and completion of a space</span></a></li>
<li class="toclevel-1 tocsection-14"><a href="#Specific_theories"><span class="tocnumber">6</span> <span class="toctext">Specific theories</span></a></li>
<li class="toclevel-1 tocsection-15"><a href="#Homotopy_hypothesis"><span class="tocnumber">7</span> <span class="toctext">Homotopy hypothesis</span></a></li>
<li class="toclevel-1 tocsection-16"><a href="#Abstract_homotopy_theory"><span class="tocnumber">8</span> <span class="toctext">Abstract homotopy theory</span></a>
<ul>
<li class="toclevel-2 tocsection-17"><a href="#Simplicial_set"><span class="tocnumber">8.1</span> <span class="toctext">Simplicial set</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-18"><a href="#See_also"><span class="tocnumber">9</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-19"><a href="#References"><span class="tocnumber">10</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-20"><a href="#Further_reading"><span class="tocnumber">11</span> <span class="toctext">Further reading</span></a></li>
<li class="toclevel-1 tocsection-21"><a href="#External_links"><span class="tocnumber">12</span> <span class="toctext">External links</span></a></li>
</ul>
</div>
<div class="mw-heading mw-heading2"><h2 id="Applications_to_other_fields_of_mathematics">Applications to other fields of mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=1" title="Edit section: Applications to other fields of mathematics" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=1" title="Edit section's source code: Applications to other fields of mathematics"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>Besides algebraic topology, the theory has also been used in other areas of mathematics such as:
</p>
<ul><li><a href="/https/en.wikipedia.org/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a> (e.g., <a href="/https/en.wikipedia.org/wiki/A1_homotopy_theory" class="mw-redirect" title="A1 homotopy theory">A<sup>1</sup> homotopy theory</a>)</li>
<li><a href="/https/en.wikipedia.org/wiki/Category_theory" title="Category theory">Category theory</a> (specifically the study of <a href="/https/en.wikipedia.org/wiki/Higher_category_theory" title="Higher category theory">higher categories</a>)</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Concepts">Concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=2" title="Edit section: Concepts" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=2" title="Edit section's source code: Concepts"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<div class="mw-heading mw-heading3"><h3 id="Spaces_and_maps">Spaces and maps</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=3" title="Edit section: Spaces and maps" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=3" title="Edit section's source code: Spaces and maps"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>In homotopy theory and algebraic topology, the word "space" denotes a <a href="/https/en.wikipedia.org/wiki/Topological_space" title="Topological space">topological space</a>. In order to avoid <a href="/https/en.wikipedia.org/wiki/Pathological_(mathematics)" title="Pathological (mathematics)">pathologies</a>, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being <a href="/https/en.wikipedia.org/wiki/Category_of_compactly_generated_weak_Hausdorff_spaces" title="Category of compactly generated weak Hausdorff spaces">compactly generated weak Hausdorff</a> or a <a href="/https/en.wikipedia.org/wiki/CW_complex" title="CW complex">CW complex</a>.
</p><p>In the same vein as above, a "<a href="/https/en.wikipedia.org/wiki/Map_(mathematics)" title="Map (mathematics)">map</a>" is a continuous function, possibly with some extra constraints.
</p><p>Often, one works with a <a href="/https/en.wikipedia.org/wiki/Pointed_space" title="Pointed space">pointed space</a>—that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserves basepoints; that is, it sends the basepoint of the domain to that of the codomain. In contrast, a free map is one which needn't preserve basepoints.
</p><p>The Cartesian product of two pointed spaces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>,</mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X,Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8705438171d938b7f59cd1bfa5b7d99b6afa5cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.787ex; height:2.509ex;" alt="{\displaystyle X,Y}"></span> are not naturally pointed. A substitute is the <a href="/https/en.wikipedia.org/wiki/Smash_product" title="Smash product">smash product</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\wedge Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>∧<!-- ∧ --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\wedge Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6edc6915b42026ef5d46c585f7e44955f2d15ecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.336ex; height:2.176ex;" alt="{\displaystyle X\wedge Y}"></span> which is characterized by the <a href="/https/en.wikipedia.org/wiki/Adjoint_functor" class="mw-redirect" title="Adjoint functor">adjoint relation</a>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Map} (X\wedge Y,Z)=\operatorname {Map} (X,\operatorname {Map} (Y,Z))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Map</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>∧<!-- ∧ --></mo>
<mi>Y</mi>
<mo>,</mo>
<mi>Z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>Map</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>,</mo>
<mi>Map</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo>,</mo>
<mi>Z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Map} (X\wedge Y,Z)=\operatorname {Map} (X,\operatorname {Map} (Y,Z))}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7c9565c7ff0768fabf7bb5cb2c57d19b905fa96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.836ex; height:2.843ex;" alt="{\displaystyle \operatorname {Map} (X\wedge Y,Z)=\operatorname {Map} (X,\operatorname {Map} (Y,Z))}"></span>,</dd></dl>
<p>that is, a smash product is an analog of a <a href="/https/en.wikipedia.org/wiki/Tensor_product" title="Tensor product">tensor product</a> in abstract algebra (see <a href="/https/en.wikipedia.org/wiki/Tensor-hom_adjunction" title="Tensor-hom adjunction">tensor-hom adjunction</a>). Explicitly, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\wedge Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>∧<!-- ∧ --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\wedge Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6edc6915b42026ef5d46c585f7e44955f2d15ecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.336ex; height:2.176ex;" alt="{\displaystyle X\wedge Y}"></span> is the quotient of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>×<!-- × --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\times Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1613c1ff4b6fbfb6c80a8da83e90ad28f0ab3483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.594ex; height:2.176ex;" alt="{\displaystyle X\times Y}"></span> by the <a href="/https/en.wikipedia.org/wiki/Wedge_sum" title="Wedge sum">wedge sum</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\vee Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>∨<!-- ∨ --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\vee Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/091f7bb09b74960d59d46cc57a297ae37aece6e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.336ex; height:2.176ex;" alt="{\displaystyle X\vee Y}"></span>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Homotopy">Homotopy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=4" title="Edit section: Homotopy" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=4" title="Edit section's source code: Homotopy"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/https/en.wikipedia.org/wiki/Homotopy" title="Homotopy">Homotopy</a></div>
<p>Let <i>I</i> denote the unit interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"></span>. A map
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:X\times I\to Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo>:</mo>
<mi>X</mi>
<mo>×<!-- × --></mo>
<mi>I</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h:X\times I\to Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f240a8efc3ead4c041585116d590cdfb0e4df75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.656ex; height:2.176ex;" alt="{\displaystyle h:X\times I\to Y}"></span></dd></dl>
<p>is called a <a href="/https/en.wikipedia.org/wiki/Homotopy" title="Homotopy">homotopy</a> from the map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c1c908b03c3f63383c7199465c7fd0b105030f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{0}}"></span> to the map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e8880a2e4243a2fe5157e574a0547ef3d5d373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{1}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{t}(x)=h(x,t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>h</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{t}(x)=h(x,t)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04b80f80ea8df70ed6fda9cd793469d578025501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.754ex; height:2.843ex;" alt="{\displaystyle h_{t}(x)=h(x,t)}"></span>. Intuitively, we may think of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> as a path from the map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c1c908b03c3f63383c7199465c7fd0b105030f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{0}}"></span> to the map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e8880a2e4243a2fe5157e574a0547ef3d5d373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{1}}"></span>. Indeed, a homotopy can be shown to be an <a href="/https/en.wikipedia.org/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>. When <i>X</i>, <i>Y</i> are pointed spaces, the maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{t}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8dbf3d8bfe322f68ff6400385578f8d78e1ba7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.165ex; height:2.509ex;" alt="{\displaystyle h_{t}}"></span> are required to preserve the basepoint and the homotopy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> is called a <a href="/https/en.wikipedia.org/w/index.php?title=Based_homotopy&action=edit&redlink=1" class="new" title="Based homotopy (page does not exist)">based homotopy</a>. A based homotopy is the same as a (based) map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\wedge I_{+}\to Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>∧<!-- ∧ --></mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msub>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\wedge I_{+}\to Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a9ae72755c47f18e95a6484b51dc4fd34ee7475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.484ex; height:2.509ex;" alt="{\displaystyle X\wedge I_{+}\to Y}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{+}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I_{+}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03b29b97c4ce7b7a7551cf9f537a5fa42c14c145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.534ex; height:2.509ex;" alt="{\displaystyle I_{+}}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> together with a disjoint basepoint.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>Given a pointed space <i>X</i> and an <a href="/https/en.wikipedia.org/wiki/Integer" title="Integer">integer</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}"></span>, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{n}X=[S^{n},X]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mi>X</mi>
<mo>=</mo>
<mo stretchy="false">[</mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>,</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \pi _{n}X=[S^{n},X]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01933c254562233f07b4a64a076e6aa6337e066a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.67ex; height:2.843ex;" alt="{\displaystyle \pi _{n}X=[S^{n},X]}"></span> be the homotopy classes of based maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n}\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{n}\to X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21881e91b66e1c3c1fd8803e144986cb52bec9ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.334ex; height:2.343ex;" alt="{\displaystyle S^{n}\to X}"></span> from a (pointed) <i>n</i>-sphere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee006452a59bf1eb29983b4412348b66517a2d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.74ex; height:2.343ex;" alt="{\displaystyle S^{n}}"></span> to <i>X</i>. As it turns out,
</p>
<ul><li>for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{n}X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \pi _{n}X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/309cf040236632f263afb0b16d69c0f0fa4f2140" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.524ex; height:2.509ex;" alt="{\displaystyle \pi _{n}X}"></span> are <a href="/https/en.wikipedia.org/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a> called <a href="/https/en.wikipedia.org/wiki/Homotopy_group" title="Homotopy group">homotopy groups</a>; in particular, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \pi _{1}X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b73939ad53b202c43c810c0d625ca2d2c10946b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.359ex; height:2.509ex;" alt="{\displaystyle \pi _{1}X}"></span> is called the <a href="/https/en.wikipedia.org/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of <i>X</i>,</li>
<li>for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{n}X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \pi _{n}X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/309cf040236632f263afb0b16d69c0f0fa4f2140" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.524ex; height:2.509ex;" alt="{\displaystyle \pi _{n}X}"></span> are <a href="/https/en.wikipedia.org/wiki/Abelian_group" title="Abelian group">abelian groups</a> by the <a href="/https/en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_argument" title="Eckmann–Hilton argument">Eckmann–Hilton argument</a>,</li>
<li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{0}X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \pi _{0}X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20676791fe5303f461e0c23f2eb5c8acd8740635" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.359ex; height:2.509ex;" alt="{\displaystyle \pi _{0}X}"></span> can be identified with the set of path-connected components in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>.</li></ul>
<p>Every group is the fundamental group of some space.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>A map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is called a <a href="/https/en.wikipedia.org/wiki/Homotopy_equivalence" class="mw-redirect" title="Homotopy equivalence">homotopy equivalence</a> if there is another map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>∘<!-- ∘ --></mo>
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\circ g}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f61ca7838709fbae07dce9c0d513770f10cfae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>∘<!-- ∘ --></mo>
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}"></span> are both homotopic to the identities. Two spaces are said to be homotopy equivalent if there is a homotopy equivalence between them. A homotopy equivalence class of spaces is then called a <a href="/https/en.wikipedia.org/wiki/Homotopy_type" class="mw-redirect" title="Homotopy type">homotopy type</a>. There is a weaker notion: a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"></span> is said to be a <a href="/https/en.wikipedia.org/wiki/Weak_homotopy_equivalence" class="mw-redirect" title="Weak homotopy equivalence">weak homotopy equivalence</a> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{*}:\pi _{n}(X)\to \pi _{n}(Y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msub>
<mo>:</mo>
<msub>
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">→<!-- → --></mo>
<msub>
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{*}:\pi _{n}(X)\to \pi _{n}(Y)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79fa1c64342226715f9327abdcb794322fb82a7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.204ex; height:2.843ex;" alt="{\displaystyle f_{*}:\pi _{n}(X)\to \pi _{n}(Y)}"></span> is an isomorphism for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}"></span> and each choice of a base point. A homotopy equivalence is a weak homotopy equivalence but the converse need not be true.
</p><p>Through the adjunction
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Map} (X\times I,Y)=\operatorname {Map} (X,\operatorname {Map} (I,Y)),\,\,h\mapsto (x\mapsto h(x,\cdot ))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Map</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>×<!-- × --></mo>
<mi>I</mi>
<mo>,</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>Map</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>,</mo>
<mi>Map</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>I</mi>
<mo>,</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mspace width="thinmathspace" />
<mspace width="thinmathspace" />
<mi>h</mi>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mi>h</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Map} (X\times I,Y)=\operatorname {Map} (X,\operatorname {Map} (I,Y)),\,\,h\mapsto (x\mapsto h(x,\cdot ))}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/527d6a194bf5615639a453ef75143bcba8cb2967" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.749ex; height:2.843ex;" alt="{\displaystyle \operatorname {Map} (X\times I,Y)=\operatorname {Map} (X,\operatorname {Map} (I,Y)),\,\,h\mapsto (x\mapsto h(x,\cdot ))}"></span>,</dd></dl>
<p>a homotopy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:X\times I\to Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo>:</mo>
<mi>X</mi>
<mo>×<!-- × --></mo>
<mi>I</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h:X\times I\to Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f240a8efc3ead4c041585116d590cdfb0e4df75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.656ex; height:2.176ex;" alt="{\displaystyle h:X\times I\to Y}"></span> is sometimes viewed as a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to Y^{I}=\operatorname {Map} (I,Y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msup>
<mo>=</mo>
<mi>Map</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>I</mi>
<mo>,</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\to Y^{I}=\operatorname {Map} (I,Y)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2fa58803125782416ad8d133cb2c740c4c2a7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.028ex; height:3.176ex;" alt="{\displaystyle X\to Y^{I}=\operatorname {Map} (I,Y)}"></span>.
</p>
<div class="mw-heading mw-heading3"><h3 id="CW_complex">CW complex</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=5" title="Edit section: CW complex" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=5" title="Edit section's source code: CW complex"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/https/en.wikipedia.org/wiki/CW_complex" title="CW complex">CW complex</a></div>
<p>A <a href="/https/en.wikipedia.org/wiki/CW_complex" title="CW complex">CW complex</a> is a space that has a filtration <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\supset \cdots \supset X^{n}\supset X^{n-1}\supset \cdots \supset X^{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>⊃<!-- ⊃ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>⊃<!-- ⊃ --></mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>⊃<!-- ⊃ --></mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo>⊃<!-- ⊃ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>⊃<!-- ⊃ --></mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\supset \cdots \supset X^{n}\supset X^{n-1}\supset \cdots \supset X^{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a171084199cd91bc40152db48e752b9f3545297" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:34.501ex; height:2.676ex;" alt="{\displaystyle X\supset \cdots \supset X^{n}\supset X^{n-1}\supset \cdots \supset X^{0}}"></span> whose union is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> and such that
</p>
<ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X^{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7ed80088727b5ba6be077bad40afbd304e84de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.676ex;" alt="{\displaystyle X^{0}}"></span> is a discrete space, called the set of 0-cells (vertices) in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>.</li>
<li>Each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X^{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/268db8293666fefd75cfb00513706171948edf09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.215ex; height:2.343ex;" alt="{\displaystyle X^{n}}"></span> is obtained by attaching several <i>n</i>-disks, <i>n</i>-cells, to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{n-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X^{n-1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98ccc911504085b636c3b947ad7534a657f18dee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.316ex; height:2.676ex;" alt="{\displaystyle X^{n-1}}"></span> via maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n-1}\to X^{n-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{n-1}\to X^{n-1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7855b6b6419f3438d4a945794d0caf166fb7ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.771ex; height:2.676ex;" alt="{\displaystyle S^{n-1}\to X^{n-1}}"></span>; i.e., the boundary of an n-disk is identified with the image of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{n-1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742bebb03fe630674b18823a59d2c75efd0066e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.676ex;" alt="{\displaystyle S^{n-1}}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{n-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X^{n-1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98ccc911504085b636c3b947ad7534a657f18dee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.316ex; height:2.676ex;" alt="{\displaystyle X^{n-1}}"></span>.</li>
<li>A subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> is open if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cap X^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo>∩<!-- ∩ --></mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U\cap X^{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60a7c5137e3642ff6d22d6cbc8d6720411cdec86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.58ex; height:2.343ex;" alt="{\displaystyle U\cap X^{n}}"></span> is open for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>.</li></ol>
<p>For example, a sphere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee006452a59bf1eb29983b4412348b66517a2d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.74ex; height:2.343ex;" alt="{\displaystyle S^{n}}"></span> has two cells: one 0-cell and one <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-cell, since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee006452a59bf1eb29983b4412348b66517a2d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.74ex; height:2.343ex;" alt="{\displaystyle S^{n}}"></span> can be obtained by collapsing the boundary <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{n-1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742bebb03fe630674b18823a59d2c75efd0066e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.676ex;" alt="{\displaystyle S^{n-1}}"></span> of the <i>n</i>-disk to a point. In general, every manifold has the homotopy type of a CW complex;<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> in fact, <a href="/https/en.wikipedia.org/wiki/Morse_theory" title="Morse theory">Morse theory</a> implies that a compact manifold has the homotopy type of a finite CW complex.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/https/en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2024)">citation needed</span></a></i>]</sup>
</p><p>Remarkably, <a href="/https/en.wikipedia.org/wiki/Whitehead%27s_theorem" class="mw-redirect" title="Whitehead's theorem">Whitehead's theorem</a> says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing.
</p><p>Another important result is the approximation theorem. First, the <a href="/https/en.wikipedia.org/wiki/Homotopy_category" title="Homotopy category">homotopy category</a> of spaces is the category where an object is a space but a morphism is the homotopy class of a map. Then
</p>
<style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style="">
<p><strong class="theorem-name"><a href="/https/en.wikipedia.org/wiki/CW_approximation" class="mw-redirect" title="CW approximation">CW approximation</a></strong><span class="theoreme-tiret"> — </span><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> There exist a functor (called the CW approximation functor)
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta :\operatorname {Ho} ({\textrm {spaces}})\to \operatorname {Ho} ({\textrm {CW}})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Θ<!-- Θ --></mi>
<mo>:</mo>
<mi>Ho</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mtext>spaces</mtext>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo stretchy="false">→<!-- → --></mo>
<mi>Ho</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mtext>CW</mtext>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Theta :\operatorname {Ho} ({\textrm {spaces}})\to \operatorname {Ho} ({\textrm {CW}})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4e5293ea913ffcabab12a1bbde2bca66dea67c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.208ex; height:2.843ex;" alt="{\displaystyle \Theta :\operatorname {Ho} ({\textrm {spaces}})\to \operatorname {Ho} ({\textrm {CW}})}"></span></dd></dl>
<p>from the homotopy category of spaces to the homotopy category of CW complexes as well as a natural transformation
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta :i\circ \Theta \to \operatorname {Id} ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo>:</mo>
<mi>i</mi>
<mo>∘<!-- ∘ --></mo>
<mi mathvariant="normal">Θ<!-- Θ --></mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Id</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta :i\circ \Theta \to \operatorname {Id} ,}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24f2f46c7e2d87fea944783b58f2b0a5a4afe449" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.226ex; height:2.509ex;" alt="{\displaystyle \theta :i\circ \Theta \to \operatorname {Id} ,}"></span></dd></dl>
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:\operatorname {Ho} ({\textrm {CW}})\hookrightarrow \operatorname {Ho} ({\textrm {spaces}})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>:</mo>
<mi>Ho</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mtext>CW</mtext>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo stretchy="false">↪<!-- ↪ --></mo>
<mi>Ho</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mtext>spaces</mtext>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i:\operatorname {Ho} ({\textrm {CW}})\hookrightarrow \operatorname {Ho} ({\textrm {spaces}})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95604963b664deaa4cb7850f63d39917b82dbd67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.495ex; height:2.843ex;" alt="{\displaystyle i:\operatorname {Ho} ({\textrm {CW}})\hookrightarrow \operatorname {Ho} ({\textrm {spaces}})}"></span>, such that each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{X}:i(\Theta (X))\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo>:</mo>
<mi>i</mi>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">Θ<!-- Θ --></mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta _{X}:i(\Theta (X))\to X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9cd1839d29889d07bd58e5fa7898ad4d21862c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.463ex; height:2.843ex;" alt="{\displaystyle \theta _{X}:i(\Theta (X))\to X}"></span> is a weak homotopy equivalence.
</p><p>Similar statements also hold for pairs and excisive triads.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>
</p>
</div>
<p>Explicitly, the above approximation functor can be defined as the composition of the <a href="/https/en.wikipedia.org/wiki/Singular_chain" class="mw-redirect" title="Singular chain">singular chain</a> functor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{*}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S_{*}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d3325410f0949bfb47c30694d58411ab83c7137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.293ex; margin-bottom: -0.379ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{*}}"></span> followed by the geometric realization functor; see <a href="#Simplicial_set">§ Simplicial set</a>.
</p><p>The above theorem justifies a common habit of working only with CW complexes. For example, given a space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, one can just define the homology of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> to the homology of the CW approximation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> (the cell structure of a CW complex determines the natural homology, the <a href="/https/en.wikipedia.org/wiki/Cellular_homology" title="Cellular homology">cellular homology</a> and that can be taken to be the homology of the complex.)
</p>
<div class="mw-heading mw-heading3"><h3 id="Cofibration_and_fibration">Cofibration and fibration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=6" title="Edit section: Cofibration and fibration" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=6" title="Edit section's source code: Cofibration and fibration"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>A map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>A</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:A\to X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23200a6d204a3980f2ba2bb829f254094c7d7e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.553ex; height:2.509ex;" alt="{\displaystyle f:A\to X}"></span> is called a <a href="/https/en.wikipedia.org/wiki/Cofibration" title="Cofibration">cofibration</a> if given:
</p>
<ol><li>A map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}:X\to Z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{0}:X\to Z}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e835a1751594e3ce73108c0101e090e9a275c493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.605ex; height:2.509ex;" alt="{\displaystyle h_{0}:X\to Z}"></span>, and</li>
<li>A homotopy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{t}:A\to Z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>:</mo>
<mi>A</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{t}:A\to Z}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba4b435ba3383dd590571704d8b7066abb868355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.91ex; height:2.509ex;" alt="{\displaystyle g_{t}:A\to Z}"></span></li></ol>
<p>such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}\circ f=g_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>∘<!-- ∘ --></mo>
<mi>f</mi>
<mo>=</mo>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{0}\circ f=g_{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb9202cdc1ab7ee1c855d64845bca53540dedfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.128ex; height:2.509ex;" alt="{\displaystyle h_{0}\circ f=g_{0}}"></span>, there exists a homotopy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{t}:X\to Z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{t}:X\to Z}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2a3b006de07338252900e446478f6a6f4955006" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.377ex; height:2.509ex;" alt="{\displaystyle h_{t}:X\to Z}"></span> that extends <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c1c908b03c3f63383c7199465c7fd0b105030f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{0}}"></span> and such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{t}\circ f=g_{t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>∘<!-- ∘ --></mo>
<mi>f</mi>
<mo>=</mo>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{t}\circ f=g_{t}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1beb124f689b567d787fbc2f98a38dfc5304f44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.672ex; height:2.509ex;" alt="{\displaystyle h_{t}\circ f=g_{t}}"></span>. An example is a <a href="/https/en.wikipedia.org/wiki/Neighborhood_deformation_retract" class="mw-redirect" title="Neighborhood deformation retract">neighborhood deformation retract</a>; that is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> contains a <a href="/https/en.wikipedia.org/wiki/Mapping_cylinder" title="Mapping cylinder">mapping cylinder</a> neighborhood of a closed subspace <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> the inclusion (e.g., a <a href="/https/en.wikipedia.org/wiki/Tubular_neighborhood" title="Tubular neighborhood">tubular neighborhood</a> of a closed submanifold).<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In fact, a cofibration can be characterized as a neighborhood deformation retract pair.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Another basic example is a <a href="/https/en.wikipedia.org/wiki/CW_pair" class="mw-redirect" title="CW pair">CW pair</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,A)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>,</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (X,A)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d53eff80e8e569a9ce3e2f20adf4e9bb17feca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.566ex; height:2.843ex;" alt="{\displaystyle (X,A)}"></span>; many often work only with CW complexes and the notion of a cofibration there is then often implicit.
</p><p>A <a href="/https/en.wikipedia.org/wiki/Fibration" title="Fibration">fibration</a> in the sense of Hurewicz is the dual notion of a cofibration: that is, a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p:X\to B}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d178a180f28c19c746685eaa14fd7071ee60eece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.554ex; height:2.509ex;" alt="{\displaystyle p:X\to B}"></span> is a fibration if given (1) a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}:Z\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>:</mo>
<mi>Z</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{0}:Z\to X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47ebad73de729f0ec29c72279e68a6e8bb2031e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.605ex; height:2.509ex;" alt="{\displaystyle h_{0}:Z\to X}"></span> and (2) a homotopy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{t}:Z\to B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>:</mo>
<mi>Z</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{t}:Z\to B}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb740774f751b4f47394a47a45ebfb297ca39771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.931ex; height:2.509ex;" alt="{\displaystyle g_{t}:Z\to B}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\circ h_{0}=g_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>∘<!-- ∘ --></mo>
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p\circ h_{0}=g_{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/977353bc68c4b8102b70459de9d30c98f9ce88ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.109ex; height:2.509ex;" alt="{\displaystyle p\circ h_{0}=g_{0}}"></span>, there exists a homotopy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{t}:Z\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>:</mo>
<mi>Z</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{t}:Z\to X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/729bc141fec1ea676628149aac00591bde0d3c80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.377ex; height:2.509ex;" alt="{\displaystyle h_{t}:Z\to X}"></span> that extends <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c1c908b03c3f63383c7199465c7fd0b105030f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{0}}"></span> and such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\circ h_{t}=g_{t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>∘<!-- ∘ --></mo>
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p\circ h_{t}=g_{t}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af02e5c5bc01896a68b31a30a1e4ebbde6701cf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.652ex; height:2.509ex;" alt="{\displaystyle p\circ h_{t}=g_{t}}"></span>.
</p><p>While a cofibration is characterized by the existence of a retract, a fibration is characterized by the existence of a section called the <a href="/https/en.wikipedia.org/w/index.php?title=Path_lifting&action=edit&redlink=1" class="new" title="Path lifting (page does not exist)">path lifting</a> as follows. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p':Np\to B^{I}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>p</mi>
<mo>′</mo>
</msup>
<mo>:</mo>
<mi>N</mi>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p':Np\to B^{I}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46fe86e289c43d3f3611a70f9e7f09d0bac36f98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.553ex; height:3.009ex;" alt="{\displaystyle p':Np\to B^{I}}"></span> be the pull-back of a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:E\to B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>:</mo>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p:E\to B}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8fc956bbe1fa571b4fdd9dafb69dcd8642a0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.35ex; height:2.509ex;" alt="{\displaystyle p:E\to B}"></span> along <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi \mapsto \chi (1):B^{I}\to B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>χ<!-- χ --></mi>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mi>χ<!-- χ --></mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>:</mo>
<msup>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \chi \mapsto \chi (1):B^{I}\to B}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4180aae06ae17fcbcb2f0615e40aa0f16aef90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.636ex; height:3.176ex;" alt="{\displaystyle \chi \mapsto \chi (1):B^{I}\to B}"></span>, called the <a href="/https/en.wikipedia.org/wiki/Mapping_path_space" class="mw-redirect" title="Mapping path space">mapping path space</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Viewing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>p</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p'}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40e623e3163571a220ed60ecb31aa78c24104b85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.944ex; height:2.843ex;" alt="{\displaystyle p'}"></span> as a homotopy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Np\times I\to B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
<mi>p</mi>
<mo>×<!-- × --></mo>
<mi>I</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Np\times I\to B}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9479387e4fdb5345c4344f559f602f5635ef771d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.623ex; height:2.509ex;" alt="{\displaystyle Np\times I\to B}"></span>, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is a fibration, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>p</mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p'}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40e623e3163571a220ed60ecb31aa78c24104b85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.944ex; height:2.843ex;" alt="{\displaystyle p'}"></span> gives a homotopy <sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s:Np\to E^{I}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo>:</mo>
<mi>N</mi>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s:Np\to E^{I}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd4738475487043baa416cd14189c6285cd91b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.729ex; height:3.009ex;" alt="{\displaystyle s:Np\to E^{I}}"></span></dd></dl>
<p>such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(e,\chi )(1)=e,\,(p^{I}\circ s)(e,\chi )=\chi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
<mo stretchy="false">(</mo>
<mi>e</mi>
<mo>,</mo>
<mi>χ<!-- χ --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>e</mi>
<mo>,</mo>
<mspace width="thinmathspace" />
<mo stretchy="false">(</mo>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msup>
<mo>∘<!-- ∘ --></mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>e</mi>
<mo>,</mo>
<mi>χ<!-- χ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>χ<!-- χ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s(e,\chi )(1)=e,\,(p^{I}\circ s)(e,\chi )=\chi }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2073d8e612a8221a4874cfb784c7b4696c0cbe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.307ex; height:3.176ex;" alt="{\displaystyle s(e,\chi )(1)=e,\,(p^{I}\circ s)(e,\chi )=\chi }"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{I}:E^{I}\to B^{I}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msup>
<mo>:</mo>
<msup>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p^{I}:E^{I}\to B^{I}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db1e5edc7958251f837450d16d05f07885c87cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.55ex; height:3.009ex;" alt="{\displaystyle p^{I}:E^{I}\to B^{I}}"></span> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> This <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> is called the path lifting associated to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>. Conversely, if there is a path lifting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is a fibration as a required homotopy is obtained via <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>.
</p><p>A basic example of a fibration is a <a href="/https/en.wikipedia.org/wiki/Covering_map" class="mw-redirect" title="Covering map">covering map</a> since it comes a unique path lifting. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> is a <a href="/https/en.wikipedia.org/wiki/Principal_bundle" title="Principal bundle">principal <i>G</i>-bundle</a> over a paracompact space, that is, a space with a <a href="/https/en.wikipedia.org/wiki/Group_action#Remarkable_properties_of_actions" title="Group action">free and transitive</a> (topological) <a href="/https/en.wikipedia.org/wiki/Group_action" title="Group action">group action</a> of a (<a href="/https/en.wikipedia.org/wiki/Topological_group" title="Topological group">topological</a>) group, then the projection map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:E\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>:</mo>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p:E\to X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5c4236298a8edc5123246d49d3a8c21107f0cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.566ex; height:2.509ex;" alt="{\displaystyle p:E\to X}"></span> is an example of a fibration, because a Hurewicz fibration can be checked locally on a paracompact space.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>
</p><p>There are also based versions of a cofibration and a fibration (namely, the maps are required to be based).<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Lifting_property">Lifting property</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=7" title="Edit section: Lifting property" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=7" title="Edit section's source code: Lifting property"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>A pair of maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:A\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>:</mo>
<mi>A</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i:A\to X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97059160d1002162c22e1f2f5e4c2aee2afaf629" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.077ex; height:2.176ex;" alt="{\displaystyle i:A\to X}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:E\to B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>:</mo>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p:E\to B}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8fc956bbe1fa571b4fdd9dafb69dcd8642a0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.35ex; height:2.509ex;" alt="{\displaystyle p:E\to B}"></span> is said to satisfy the <a href="/https/en.wikipedia.org/wiki/Lifting_property" title="Lifting property">lifting property</a> <sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> if for each commutative square diagram
</p>
<dl><dd><span typeof="mw:File/Frameless"><a href="/https/en.wikipedia.org/wiki/File:Lifting_property_diagram.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Lifting_property_diagram.png/150px-Lifting_property_diagram.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Lifting_property_diagram.png/225px-Lifting_property_diagram.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Lifting_property_diagram.png/300px-Lifting_property_diagram.png 2x" data-file-width="2560" data-file-height="2560" /></a></span></dd></dl>
<p>there is a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>λ<!-- λ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> that makes the above diagram still commute. (The notion originates in the theory of <a href="/https/en.wikipedia.org/wiki/Model_category" title="Model category">model categories</a>.)
</p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {c}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">c</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {c}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21924b960341255be18e538e51404718f29cbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {c}}}"></span> be a class of maps. Then a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:E\to B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>:</mo>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p:E\to B}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8fc956bbe1fa571b4fdd9dafb69dcd8642a0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.35ex; height:2.509ex;" alt="{\displaystyle p:E\to B}"></span> is said to satisfy the <a href="/https/en.wikipedia.org/wiki/Right_lifting_property" class="mw-redirect" title="Right lifting property">right lifting property</a> or the RLP if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> satisfies the above lifting property for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {c}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">c</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {c}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21924b960341255be18e538e51404718f29cbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {c}}}"></span>. Similarly, a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:A\to X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>:</mo>
<mi>A</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i:A\to X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97059160d1002162c22e1f2f5e4c2aee2afaf629" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.077ex; height:2.176ex;" alt="{\displaystyle i:A\to X}"></span> is said to satisfy the <a href="/https/en.wikipedia.org/wiki/Left_lifting_property" class="mw-redirect" title="Left lifting property">left lifting property</a> or the LLP if it satisfies the lifting property for all the maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {c}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">c</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {c}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21924b960341255be18e538e51404718f29cbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {c}}}"></span>.
</p><p>For example, a Hurewicz fibration is exactly a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:E\to B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>:</mo>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p:E\to B}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8fc956bbe1fa571b4fdd9dafb69dcd8642a0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.35ex; height:2.509ex;" alt="{\displaystyle p:E\to B}"></span> that satisfies the RLP for the inclusions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{0}:A\to A\times I}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>:</mo>
<mi>A</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>A</mi>
<mo>×<!-- × --></mo>
<mi>I</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i_{0}:A\to A\times I}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c231234a8d2dc7a24716e86de39aba7534b7146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.906ex; height:2.509ex;" alt="{\displaystyle i_{0}:A\to A\times I}"></span>. A <a href="/https/en.wikipedia.org/wiki/Serre_fibration" class="mw-redirect" title="Serre fibration">Serre fibration</a> is a map satisfying the RLP for the inclusions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:S^{n-1}\to D^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>:</mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>D</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i:S^{n-1}\to D^{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/796199b301dc6c0315beaa3695eecb9db6a522c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.337ex; height:2.676ex;" alt="{\displaystyle i:S^{n-1}\to D^{n}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{-1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b27b44abb494176cfeb76818591f178f034f28e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.854ex; height:2.676ex;" alt="{\displaystyle S^{-1}}"></span> is the empty set. A Hurewicz fibration is a Serre fibration and the converse holds for CW complexes.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>
</p><p>On the other hand, a cofibration is a map satisfying the LLP for evaluation maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:B^{I}\to B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>:</mo>
<msup>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>I</mi>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p:B^{I}\to B}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f39eab96c3f2c05a3a3ec7446bad679ab9db10a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.399ex; height:3.009ex;" alt="{\displaystyle p:B^{I}\to B}"></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Loop_and_suspension">Loop and suspension</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=8" title="Edit section: Loop and suspension" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=8" title="Edit section's source code: Loop and suspension"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>On the category of pointed spaces, there are two important functors: the <a href="/https/en.wikipedia.org/wiki/Loop_functor" class="mw-redirect" title="Loop functor">loop functor</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> and the (reduced) <a href="/https/en.wikipedia.org/wiki/Suspension_functor" class="mw-redirect" title="Suspension functor">suspension functor</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Σ<!-- Σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"></span>, which are in the <a href="/https/en.wikipedia.org/wiki/Adjoint_functor" class="mw-redirect" title="Adjoint functor">adjoint relation</a>. Precisely, they are defined as<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>
</p>
<ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega X=\operatorname {Map} (S^{1},X)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mi>X</mi>
<mo>=</mo>
<mi>Map</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
<mo>,</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega X=\operatorname {Map} (S^{1},X)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be19b146eaa63d0d4f30affcbfc4c5232afb1a65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.742ex; height:3.176ex;" alt="{\displaystyle \Omega X=\operatorname {Map} (S^{1},X)}"></span>, and</li>
<li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma X=X\wedge S^{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Σ<!-- Σ --></mi>
<mi>X</mi>
<mo>=</mo>
<mi>X</mi>
<mo>∧<!-- ∧ --></mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Sigma X=X\wedge S^{1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a4d0c76d6820007190b14819cdab1dd7f873e1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.895ex; height:2.676ex;" alt="{\displaystyle \Sigma X=X\wedge S^{1}}"></span>.</li></ul>
<p>Because of the adjoint relation between a smash product and a mapping space, we have:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Map} (\Sigma X,Y)=\operatorname {Map} (X,\Omega Y).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Map</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">Σ<!-- Σ --></mi>
<mi>X</mi>
<mo>,</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>Map</mi>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>,</mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Map} (\Sigma X,Y)=\operatorname {Map} (X,\Omega Y).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8aa20d7e37fe5ff4b62ddd519e3533955eee634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.466ex; height:2.843ex;" alt="{\displaystyle \operatorname {Map} (\Sigma X,Y)=\operatorname {Map} (X,\Omega Y).}"></span></dd></dl>
<p>These functors are used to construct <a href="/https/en.wikipedia.org/wiki/Fiber_sequence" class="mw-redirect" title="Fiber sequence">fiber sequences</a> and <a href="/https/en.wikipedia.org/wiki/Cofiber_sequence" class="mw-redirect" title="Cofiber sequence">cofiber sequences</a>. Namely, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"></span> is a map, the fiber sequence generated by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is the exact sequence<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdots \to \Omega ^{2}Ff\to \Omega ^{2}X\to \Omega ^{2}Y\to \Omega Ff\to \Omega X\to \Omega Y\to Ff\to X\to Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>⋯<!-- ⋯ --></mo>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>F</mi>
<mi>f</mi>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>Y</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mi>F</mi>
<mi>f</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mi>Y</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>F</mi>
<mi>f</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cdots \to \Omega ^{2}Ff\to \Omega ^{2}X\to \Omega ^{2}Y\to \Omega Ff\to \Omega X\to \Omega Y\to Ff\to X\to Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d78dc87443b5998f559fbc4300a9375493b2b3ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:68.799ex; height:3.009ex;" alt="{\displaystyle \cdots \to \Omega ^{2}Ff\to \Omega ^{2}X\to \Omega ^{2}Y\to \Omega Ff\to \Omega X\to \Omega Y\to Ff\to X\to Y}"></span></dd></dl>
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ff}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Ff}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b8098a4dce98b91535c1705a918f091d8e16ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.019ex; height:2.509ex;" alt="{\displaystyle Ff}"></span> is the <a href="/https/en.wikipedia.org/wiki/Homotopy_fiber" title="Homotopy fiber">homotopy fiber</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>; i.e., a fiber obtained after replacing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> by a (based) fibration. The cofibration sequence generated by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to Y\to Cf\to \Sigma X\to \cdots ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Y</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>C</mi>
<mi>f</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">Σ<!-- Σ --></mi>
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\to Y\to Cf\to \Sigma X\to \cdots ,}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/865ccf092466a6defe1fb8bb73a560c4310a5c1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.67ex; height:2.509ex;" alt="{\displaystyle X\to Y\to Cf\to \Sigma X\to \cdots ,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Cf}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Cf}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/333df6a3322fb29fd7198e3244dd70ab570f2d37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.045ex; height:2.509ex;" alt="{\displaystyle Cf}"></span> is the homotooy cofiber of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> constructed like a homotopy fiber (use a quotient instead of a fiber.)
</p><p>The functors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ,\Sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>,</mo>
<mi mathvariant="normal">Σ<!-- Σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega ,\Sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dd1af4cb0f984e5ecd9ff1b5ba4de669f4798e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.39ex; height:2.509ex;" alt="{\displaystyle \Omega ,\Sigma }"></span> restrict to the category of CW complexes in the following weak sense: a theorem of Milnor says that if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> has the homotopy type of a CW complex, then so does its loop space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b372331db039fbac1df6fa94dbc87af06a95cfa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.658ex; height:2.176ex;" alt="{\displaystyle \Omega X}"></span>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Classifying_spaces_and_homotopy_operations">Classifying spaces and homotopy operations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=9" title="Edit section: Classifying spaces and homotopy operations" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=9" title="Edit section's source code: Classifying spaces and homotopy operations"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>Given a topological group <i>G</i>, the <a href="/https/en.wikipedia.org/wiki/Classifying_space" title="Classifying space">classifying space</a> for <a href="/https/en.wikipedia.org/wiki/Principal_bundle" title="Principal bundle">principal <i>G</i>-bundles</a> ("the" up to equivalence) is a space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BG}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mi>G</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle BG}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/773ca20b2080cb3766062a5451a01d2220e9b067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.591ex; height:2.176ex;" alt="{\displaystyle BG}"></span> such that, for each space <i>X</i>,
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,BG]=}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo>,</mo>
<mi>B</mi>
<mi>G</mi>
<mo stretchy="false">]</mo>
<mo>=</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [X,BG]=}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b556f7b12eeaf4340043c849e9e196eeaf19d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.352ex; height:2.843ex;" alt="{\displaystyle [X,BG]=}"></span> {principal <i>G</i>-bundle on <i>X</i>} / ~ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ,\,\,[f]\mapsto [f^{*}EG]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>,</mo>
<mspace width="thinmathspace" />
<mspace width="thinmathspace" />
<mo stretchy="false">[</mo>
<mi>f</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mo stretchy="false">[</mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msup>
<mi>E</mi>
<mi>G</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle ,\,\,[f]\mapsto [f^{*}EG]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/281b96260477a392b2ccad083a9146d743556f90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.265ex; height:2.843ex;" alt="{\displaystyle ,\,\,[f]\mapsto [f^{*}EG]}"></span></dd></dl>
<p>where
</p>
<ul><li>the left-hand side is the set of homotopy classes of maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to BG}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>B</mi>
<mi>G</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\to BG}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ab00a9aeca38e80f4567f05918c4418858784a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.185ex; height:2.176ex;" alt="{\displaystyle X\to BG}"></span>,</li>
<li>~ refers isomorphism of bundles, and</li>
<li>= is given by pulling-back the distinguished bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle EG}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
<mi>G</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle EG}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7ebfb4d29c3c955d9a1cabc6f2305b94c8bbce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.602ex; height:2.176ex;" alt="{\displaystyle EG}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BG}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mi>G</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle BG}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/773ca20b2080cb3766062a5451a01d2220e9b067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.591ex; height:2.176ex;" alt="{\displaystyle BG}"></span> (called universal bundle) along a map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to BG}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>B</mi>
<mi>G</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\to BG}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ab00a9aeca38e80f4567f05918c4418858784a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.185ex; height:2.176ex;" alt="{\displaystyle X\to BG}"></span>.</li></ul>
<p><a href="/https/en.wikipedia.org/wiki/Brown%27s_representability_theorem" title="Brown's representability theorem">Brown's representability theorem</a> guarantees the existence of classifying spaces.
</p>
<div class="mw-heading mw-heading3"><h3 id="Spectrum_and_generalized_cohomology">Spectrum and generalized cohomology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=10" title="Edit section: Spectrum and generalized cohomology" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=10" title="Edit section's source code: Spectrum and generalized cohomology"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/https/en.wikipedia.org/wiki/Spectrum_(algebraic_topology)" class="mw-redirect" title="Spectrum (algebraic topology)">Spectrum (algebraic topology)</a> and <a href="/https/en.wikipedia.org/wiki/Generalized_cohomology" class="mw-redirect" title="Generalized cohomology">Generalized cohomology</a></div>
<p>The idea that a classifying space classifies principal bundles can be pushed further. For example, one might try to classify cohomology classes: given an <a href="/https/en.wikipedia.org/wiki/Abelian_group" title="Abelian group">abelian group</a> <i>A</i> (such as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>),
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X,K(A,n)]=\operatorname {H} ^{n}(X;A)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo>,</mo>
<mi>K</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>,</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
<mo>=</mo>
<msup>
<mi mathvariant="normal">H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>⁡<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>;</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [X,K(A,n)]=\operatorname {H} ^{n}(X;A)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4751d41be66b271a292e8bf33341bd2829febc6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.981ex; height:2.843ex;" alt="{\displaystyle [X,K(A,n)]=\operatorname {H} ^{n}(X;A)}"></span></dd></dl>
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(A,n)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>,</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K(A,n)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa5cc0602bbce5d12b4b2b6e1444efe75ae51e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.047ex; height:2.843ex;" alt="{\displaystyle K(A,n)}"></span> is the <a href="/https/en.wikipedia.org/wiki/Eilenberg%E2%80%93MacLane_space" title="Eilenberg–MacLane space">Eilenberg–MacLane space</a>. The above equation leads to the notion of a generalized cohomology theory; i.e., a <a href="/https/en.wikipedia.org/wiki/Contravariant_functor" class="mw-redirect" title="Contravariant functor">contravariant functor</a> from the category of spaces to the <a href="/https/en.wikipedia.org/wiki/Category_of_abelian_groups" title="Category of abelian groups">category of abelian groups</a> that satisfies the axioms generalizing ordinary cohomology theory. As it turns out, such a functor may not be <a href="/https/en.wikipedia.org/wiki/Representable_functor" title="Representable functor">representable</a> by a space but it can always be represented by a sequence of (pointed) spaces with structure maps called a spectrum. In other words, to give a generalized cohomology theory is to give a spectrum. A <a href="/https/en.wikipedia.org/wiki/K-theory" title="K-theory">K-theory</a> is an example of a generalized cohomology theory.
</p><p>A basic example of a spectrum is a <a href="/https/en.wikipedia.org/wiki/Sphere_spectrum" title="Sphere spectrum">sphere spectrum</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{0}\to S^{1}\to S^{2}\to \cdots }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mo>⋯<!-- ⋯ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S^{0}\to S^{1}\to S^{2}\to \cdots }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/674deaec2e7c73fd6545abf9d04ec2d8222eab05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.293ex; height:2.676ex;" alt="{\displaystyle S^{0}\to S^{1}\to S^{2}\to \cdots }"></span>
</p>
<div class="mw-heading mw-heading2"><h2 id="Key_theorems">Key theorems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=11" title="Edit section: Key theorems" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=11" title="Edit section's source code: Key theorems"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<ul><li><a href="/https/en.wikipedia.org/wiki/Seifert%E2%80%93van_Kampen_theorem" class="mw-redirect" title="Seifert–van Kampen theorem">Seifert–van Kampen theorem</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Homotopy_excision_theorem" title="Homotopy excision theorem">Homotopy excision theorem</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Freudenthal_suspension_theorem" title="Freudenthal suspension theorem">Freudenthal suspension theorem</a> (a corollary of the excision theorem)</li>
<li><a href="/https/en.wikipedia.org/wiki/Landweber_exact_functor_theorem" title="Landweber exact functor theorem">Landweber exact functor theorem</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Dold%E2%80%93Kan_correspondence" title="Dold–Kan correspondence">Dold–Kan correspondence</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_argument" title="Eckmann–Hilton argument">Eckmann–Hilton argument</a> - this shows for instance higher homotopy groups are <a href="/https/en.wikipedia.org/wiki/Abelian_group" title="Abelian group">abelian</a>.</li>
<li><a href="/https/en.wikipedia.org/wiki/Universal_coefficient_theorem" title="Universal coefficient theorem">Universal coefficient theorem</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Dold%E2%80%93Thom_theorem" title="Dold–Thom theorem">Dold–Thom theorem</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Obstruction_theory_and_characteristic_class">Obstruction theory and characteristic class</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=12" title="Edit section: Obstruction theory and characteristic class" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=12" title="Edit section's source code: Obstruction theory and characteristic class"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
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<p>See also: <a href="/https/en.wikipedia.org/wiki/Characteristic_class" title="Characteristic class">Characteristic class</a>, <a href="/https/en.wikipedia.org/wiki/Postnikov_tower" class="mw-redirect" title="Postnikov tower">Postnikov tower</a>, <a href="/https/en.wikipedia.org/wiki/Whitehead_torsion" title="Whitehead torsion">Whitehead torsion</a>
</p>
<div class="mw-heading mw-heading2"><h2 id="Localization_and_completion_of_a_space">Localization and completion of a space</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=13" title="Edit section: Localization and completion of a space" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=13" title="Edit section's source code: Localization and completion of a space"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
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<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/https/en.wikipedia.org/wiki/Localization_of_a_topological_space" title="Localization of a topological space">Localization of a topological space</a></div>
<div class="mw-heading mw-heading2"><h2 id="Specific_theories">Specific theories</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=14" title="Edit section: Specific theories" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=14" title="Edit section's source code: Specific theories"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>There are several specific theories
</p>
<ul><li><a href="/https/en.wikipedia.org/wiki/Simple_homotopy_theory" title="Simple homotopy theory">simple homotopy theory</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Stable_homotopy_theory" title="Stable homotopy theory">stable homotopy theory</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Chromatic_homotopy_theory" title="Chromatic homotopy theory">chromatic homotopy theory</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Rational_homotopy_theory" title="Rational homotopy theory">rational homotopy theory</a></li>
<li><a href="/https/en.wikipedia.org/w/index.php?title=P-adic_homotopy_theory&action=edit&redlink=1" class="new" title="P-adic homotopy theory (page does not exist)">p-adic homotopy theory</a></li>
<li><a href="/https/en.wikipedia.org/w/index.php?title=Equivariant_homotopy_theory&action=edit&redlink=1" class="new" title="Equivariant homotopy theory (page does not exist)">equivariant homotopy theory</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Homotopy_hypothesis">Homotopy hypothesis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=15" title="Edit section: Homotopy hypothesis" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=15" title="Edit section's source code: Homotopy hypothesis"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/https/en.wikipedia.org/wiki/Homotopy_hypothesis" title="Homotopy hypothesis">Homotopy hypothesis</a></div>
<p>One of the basic questions in the foundations of homotopy theory is the nature of a space. The <a href="/https/en.wikipedia.org/wiki/Homotopy_hypothesis" title="Homotopy hypothesis">homotopy hypothesis</a> asks whether a space is something fundamentally algebraic.
</p><p>If one prefers to work with a space instead of a pointed space, there is the notion of a <a href="/https/en.wikipedia.org/wiki/Fundamental_groupoid" title="Fundamental groupoid">fundamental groupoid</a> (and higher variants): by definition, the fundamental groupoid of a space <i>X</i> is the <a href="/https/en.wikipedia.org/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> where the <a href="/https/en.wikipedia.org/wiki/Object_(category_theory)" class="mw-redirect" title="Object (category theory)">objects</a> are the points of <i>X</i> and the <a href="/https/en.wikipedia.org/wiki/Morphism" title="Morphism">morphisms</a> are paths.
</p>
<div class="mw-heading mw-heading2"><h2 id="Abstract_homotopy_theory">Abstract homotopy theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=16" title="Edit section: Abstract homotopy theory" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=16" title="Edit section's source code: Abstract homotopy theory"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>Abstract homotopy theory is an axiomatic approach to homotopy theory. Such axiomatization is useful for non-traditional applications of homotopy theory. One approach to axiomatization is by Quillen's <a href="/https/en.wikipedia.org/wiki/Model_category" title="Model category">model categories</a>. A model category is a category with a choice of three classes of maps called weak equivalences, cofibrations and fibrations, subject to the axioms that are reminiscent of facts in algebraic topology. For example, the category of (reasonable) topological spaces has a structure of a model category where a weak equivalence is a weak homotopy equivalence, a cofibration a certain retract and a fibration a Serre fibration.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Another example is the category of non-negatively graded chain complexes over a fixed base ring.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>
</p><p>See also: <a href="/https/en.wikipedia.org/wiki/Algebraic_homotopy" title="Algebraic homotopy">Algebraic homotopy</a>
</p>
<div class="mw-heading mw-heading3"><h3 id="Simplicial_set">Simplicial set</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=17" title="Edit section: Simplicial set" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=17" title="Edit section's source code: Simplicial set"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/https/en.wikipedia.org/wiki/Simplicial_set" title="Simplicial set">Simplicial set</a> and <a href="/https/en.wikipedia.org/wiki/Simplicial_homotopy_theory" class="mw-redirect" title="Simplicial homotopy theory">simplicial homotopy theory</a></div>
<p>A <a href="/https/en.wikipedia.org/wiki/Simplicial_set" title="Simplicial set">simplicial set</a> is an abstract generalization of a <a href="/https/en.wikipedia.org/wiki/Simplicial_complex" title="Simplicial complex">simplicial complex</a> and can play a role of a "space" in some sense. Despite the name, it is not a set but is a sequence of sets together with the certain maps (face and degeneracy) between those sets.
</p><p>For example, given a space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, for each integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}"></span>, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S_{n}X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52fba2d5367b62ae540877131ffde8925f0f5532" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.623ex; height:2.509ex;" alt="{\displaystyle S_{n}X}"></span> be the set of all maps from the <i>n</i>-simplex to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. Then the sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S_{n}X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52fba2d5367b62ae540877131ffde8925f0f5532" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.623ex; height:2.509ex;" alt="{\displaystyle S_{n}X}"></span> of sets is a simplicial set.<sup id="cite_ref-May_simplicial_21-0" class="reference"><a href="#cite_note-May_simplicial-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Each simplicial set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\{K_{n}\}_{n\geq 0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msub>
<mo fence="false" stretchy="false">}</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K=\{K_{n}\}_{n\geq 0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b86aba796b67931dbedb9089b20934097a8d9eca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14ex; height:2.843ex;" alt="{\displaystyle K=\{K_{n}\}_{n\geq 0}}"></span> has a naturally associated chain complex and the homology of that chain complex is the homology of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>. The <a href="/https/en.wikipedia.org/wiki/Singular_homology" title="Singular homology">singular homology</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is precisely the homology of the simplicial set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{*}X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
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</msub>
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S_{*}X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6184c0414df27e52aaf2c0412771b6d6ea60fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.293ex; margin-bottom: -0.379ex; width:4.459ex; height:2.509ex;" alt="{\displaystyle S_{*}X}"></span>. Also, the <a href="/https/en.wikipedia.org/wiki/Simplicial_set#Geometric_realization" title="Simplicial set">geometric realization</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mo>⋅<!-- ⋅ --></mo>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6c0e7be2b4c1e2722d4d30ca14e4f84d6be3e3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.347ex; height:2.843ex;" alt="{\displaystyle X\mapsto |S_{*}X|}"></span> is precisely the CW approximation functor.
</p><p>Another important example is a category or more precisely the <a href="/https/en.wikipedia.org/wiki/Nerve_of_a_category" class="mw-redirect" title="Nerve of a category">nerve of a category</a>, which is a simplicial set. In fact, a simplicial set is the nerve of some category if and only if it satisfies the <a href="/https/en.wikipedia.org/wiki/Segal_condition" class="mw-redirect" title="Segal condition">Segal conditions</a> (a theorem of Grothendieck). Each category is completely determined by its nerve. In this way, a category can be viewed as a special kind of a simplicial set, and this observation is used to generalize a category. Namely, an <a href="/https/en.wikipedia.org/wiki/Infinity_category" class="mw-redirect" title="Infinity category"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span>-groupoid</a> is defined as particular kinds of simplicial sets.
</p><p>Since simplicial sets are sort of abstract spaces (if not topological spaces), it is possible to develop the homotopy theory on them, which is called the <a href="/https/en.wikipedia.org/wiki/Simplicial_homotopy_theory" class="mw-redirect" title="Simplicial homotopy theory">simplicial homotopy theory</a>.<sup id="cite_ref-May_simplicial_21-1" class="reference"><a href="#cite_note-May_simplicial-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=18" title="Edit section: See also" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=18" title="Edit section's source code: See also"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<ul><li><a href="/https/en.wikipedia.org/wiki/Highly_structured_ring_spectrum" title="Highly structured ring spectrum">Highly structured ring spectrum</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Homotopy_type_theory" title="Homotopy type theory">Homotopy type theory</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Pursuing_Stacks" title="Pursuing Stacks">Pursuing Stacks</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Shape_theory_(mathematics)" title="Shape theory (mathematics)">Shape theory</a></li>
<li><a href="/https/en.wikipedia.org/wiki/Moduli_stack_of_formal_group_laws" title="Moduli stack of formal group laws">Moduli stack of formal group laws</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=19" title="Edit section: References" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=19" title="Edit section's source code: References"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist">
<div class="mw-references-wrap mw-references-columns"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 8. § 3.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFMay (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch 4. § 5.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFMay (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFMilnor1959">Milnor 1959</a>, Corollary 1. NB: "second countable" implies "separable".</span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 10., § 5<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFMay (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 10., § 6<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFMay (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
</li>
<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 10., § 7<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFMay (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
</li>
<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFHatcher">Hatcher</a>, Example 0.15.</span>
</li>
<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch 6. § 4.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFMay (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
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<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Some authors use <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi \mapsto \chi (0)}">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/175d4ac82c0ccf05ffd0030a792a1677006670de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.496ex; height:2.843ex;" alt="{\displaystyle \chi \mapsto \chi (0)}"></span>. The definition here is from <a href="#CITEREFMay">May</a>, Ch. 8., § 5.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFMay (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
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<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 7., § 2.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFMay (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
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<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
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</li>
<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 7., § 4.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFMay (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
</li>
<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch 8. § 3. and § 5.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFMay (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
</li>
<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFMayPonto">May & Ponto</a>, Definition 14.1.5.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFMayPonto (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
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<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="https://ncatlab.org/nlab/show/a+Serre+fibration+between+CW-complexes+is+a+Hurewicz+fibration">https://ncatlab.org/nlab/show/a+Serre+fibration+between+CW-complexes+is+a+Hurewicz+fibration</a></span>
</li>
<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 8, § 2.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFMay (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
</li>
<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 8, § 6.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFMay (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
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<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a href="#CITEREFMilnor1959">Milnor 1959</a>, Theorem 3.</span>
</li>
<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a href="#CITEREFDwyerSpalinski">Dwyer & Spalinski</a>, Example 3.5.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFDwyerSpalinski (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
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<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a href="#CITEREFDwyerSpalinski">Dwyer & Spalinski</a>, Example 3.7.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFDwyerSpalinski (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
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<li id="cite_note-May_simplicial-21"><span class="mw-cite-backlink">^ <a href="#cite_ref-May_simplicial_21-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-May_simplicial_21-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFMay">May</a>, Ch. 16, § 4.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFMay (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
</li>
</ol></div></div>
<ul><li>May, J. <a rel="nofollow" class="external text" href="http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf">A Concise Course in Algebraic Topology</a></li>
<li>J. P. May and K. Ponto, More Concise Algebraic Topology: Localization, completion, and model categories</li>
<li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFGeorge_William_Whitehead1978" class="citation book cs1"><a href="/https/en.wikipedia.org/wiki/George_W._Whitehead" title="George W. Whitehead">George William Whitehead</a> (1978). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wlrvAAAAMAAJ"><i>Elements of homotopy theory</i></a>. Graduate Texts in Mathematics. Vol. 61 (3rd ed.). New York-Berlin: Springer-Verlag. pp. xxi+744. <a href="/https/en.wikipedia.org/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/https/en.wikipedia.org/wiki/Special:BookSources/978-0-387-90336-1" title="Special:BookSources/978-0-387-90336-1"><bdi>978-0-387-90336-1</bdi></a>. <a href="/https/en.wikipedia.org/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0516508">0516508</a><span class="reference-accessdate">. Retrieved <span class="nowrap">September 6,</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+homotopy+theory&rft.place=New+York-Berlin&rft.series=Graduate+Texts+in+Mathematics&rft.pages=xxi%2B744&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=1978&rft.isbn=978-0-387-90336-1&rft_id=https%3A%2F%2Ffanyv88.com%3A443%2Fhttps%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0516508%23id-name%3DMR&rft.au=George+William+Whitehead&rft_id=https%3A%2F%2Ffanyv88.com%3A443%2Fhttps%2Fbooks.google.com%2Fbooks%3Fid%3DwlrvAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li>
<li>Ronald Brown, <i><a rel="nofollow" class="external text" href="http://arquivo.pt/wayback/20160514115224/http://www.bangor.ac.uk/r.brown/topgpds.html">Topology and groupoids</a></i> (2006) Booksurge LLC <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/https/en.wikipedia.org/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/https/en.wikipedia.org/wiki/Special:BookSources/1-4196-2722-8" title="Special:BookSources/1-4196-2722-8">1-4196-2722-8</a>.</li>
<li><a rel="nofollow" class="external free" href="https://ncatlab.org/nlab/show/homotopical+algebra">https://ncatlab.org/nlab/show/homotopical+algebra</a></li>
<li>Homotopy Theories and Model Categories by W.G. Dwyer and J. Spalinski in <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xoM5DxQZihQC&printsec=copyright#v=onepage&q&f=false">Handbook of Algebraic Topology</a> edited by I.M. James</li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHatcher" class="citation web cs1">Hatcher, Allen. <a rel="nofollow" class="external text" href="http://www.math.cornell.edu/~hatcher/AT/ATpage.html">"Algebraic topology"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Algebraic+topology&rft.aulast=Hatcher&rft.aufirst=Allen&rft_id=https%3A%2F%2Ffanyv88.com%3A443%2Fhttp%2Fwww.math.cornell.edu%2F~hatcher%2FAT%2FATpage.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilnor1959" class="citation journal cs1">Milnor, John (1959). <a rel="nofollow" class="external text" href="https://www.semanticscholar.org/paper/On-spaces-having-the-homotopy-type-of-a-CW-complex-Milnor/905bb7242d4e2b7b7e168d12718b6595c98e98d9">"On spaces having the homotopy type of 𝐶𝑊-complex"</a>. <i>Transactions of the American Mathematical Society</i>. <b>90</b> (2): 272–280. <a href="/https/en.wikipedia.org/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9947-1959-0100267-4">10.1090/S0002-9947-1959-0100267-4</a>. <a href="/https/en.wikipedia.org/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9947">0002-9947</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=On+spaces+having+the+homotopy+type+of+%F0%9D%90%B6%F0%9D%91%8A-complex&rft.volume=90&rft.issue=2&rft.pages=272-280&rft.date=1959&rft_id=info%3Adoi%2F10.1090%2FS0002-9947-1959-0100267-4&rft.issn=0002-9947&rft.aulast=Milnor&rft.aufirst=John&rft_id=https%3A%2F%2Ffanyv88.com%3A443%2Fhttps%2Fwww.semanticscholar.org%2Fpaper%2FOn-spaces-having-the-homotopy-type-of-a-CW-complex-Milnor%2F905bb7242d4e2b7b7e168d12718b6595c98e98d9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHomotopy+theory" class="Z3988"></span></li>
<li>Edwin Spanier, Algebraic topology</li>
<li>Dennis Sullivan. Genetics of homotopy theory and the Adams conjecture. Ann. of Math. (2), 100:1–79, 1974.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=20" title="Edit section: Further reading" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=20" title="Edit section's source code: Further reading"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<ul><li><a rel="nofollow" class="external text" href="http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf">Cisinski's notes</a></li>
<li><a rel="nofollow" class="external free" href="http://ncatlab.org/nlab/files/Abstract-Homotopy.pdf">http://ncatlab.org/nlab/files/Abstract-Homotopy.pdf</a></li>
<li><a rel="nofollow" class="external text" href="https://uregina.ca/~franklam/Math527/Math527.html">Math 527 - Homotopy Theory Spring 2013, Section F1</a>, lectures by Martin Frankland</li>
<li>D. Quillen, Homotopical algebra, Lectures Notes in Math. vol. 43, Springer Verlag, 1967.</li>
<li><a rel="nofollow" class="external free" href="https://ncatlab.org/nlab/show/homotopy+theory">https://ncatlab.org/nlab/show/homotopy+theory</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&veaction=edit&section=21" title="Edit section: External links" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Homotopy_theory&action=edit&section=21" title="Edit section's source code: External links"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
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