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'{{Short description|Standard that diagrams must satisfy up to isomorphism}} In [[mathematics]], specifically in [[homotopy theory]] and [[higher category theory|(higher) category theory]], '''coherency''' is the standard that equalities or diagrams must satisfy when they hold "[[up to]] [[homotopy]]" or "up to [[isomorphism]]". The adjectives such as "pseudo-" and "lax-" are used to refer to the fact equalities are weakened in coherent ways; e.g., [[pseudo-functor]], [[pseudoalgebra]]. == Coherent isomorphism == In some situations, isomorphisms need to be chosen in a coherent way. Often, this can be achieved by choosing [[canonical isomorphism]]s. But in some cases, such as [[prestack]]s, there can be several canonical isomorphisms and there might not be an obvious choice among them. In practice, coherent isomorphisms arise by weakening equalities; e.g., strict [[Associative property|associativity]] may be replaced by associativity via coherent isomorphisms. For example, via this process, one gets the notion of a [[weak 2-category]] from that of a [[strict 2-category]]. Replacing coherent isomorphisms by equalities is usually called strictification or rectification. == Coherence theorem == [[Mac Lane's coherence theorem]] states, roughly, that if diagrams of certain types [[Commutative diagram|commute]], then diagrams of all types commute.<ref>{{harvnb|Mac Lane|1978|loc=Chapter VII, Section 2}}</ref> A simple proof of that theorem can be obtained using the [[permutoassociahedron]], a [[polytope]] whose combinatorial structure appears implicitly in Mac Lane's proof.<ref>See {{harvnb|Kapranov|1993}} and {{harvnb|Reiner|Ziegler|1994}}</ref><!-- precise statement here --> There are several generalizations of Mac Lane's coherence theorem.<ref>See, for instance [https://ncatlab.org/nlab/show/coherence+theorem coherence theorem (nlab)]</ref> Each of them has the rough form that "every weak structure of some sort is equivalent to a stricter one".<ref>{{harvnb|Shulman|2012|loc=Section 1}}</ref> == Homotopy coherence == {{expand section|date=September 2019}} == See also == *[[Coherence condition]] *[[Canonical isomorphism]] == Notes == {{reflist}} == References == *{{cite journal | last1=Cordier | first1=Jean-Marc | last2=Porter | first2=Timothy | title=Homotopy coherent category theory | journal=[[Transactions of the American Mathematical Society]] | doi=10.1090/S0002-9947-97-01752-2 | doi-access=free | volume=349 | issue=1 | date=1997 | pages=1–54}} * § 5. of {{cite journal | last1=Mac Lane | first1=Saunders | authorlink1=Saunders Mac Lane | title=Topology and Logic as a Source of Algebra (Retiring Presidential Address) | journal=Bulletin of the American Mathematical Society | volume=82 | issue=1 | date=January 1976 | pages=1–40 | doi=10.1090/S0002-9904-1976-13928-6 | doi-access=free}} * {{cite book | last1=Mac Lane | first1=Saunders | authorlink1=Saunders Mac Lane | orig-year=1971 | title=Categories for the working mathematician | series=Graduate texts in mathematics | publisher=Springer-Verlag | date=1978 | doi=10.1007/978-1-4757-4721-8 | doi-access=free}} * Ch. 5 of {{cite book | last1=Kamps | first1=Klaus Heiner | last2=Porter | first2=Timothy | title=Abstract Homotopy and Simple Homotopy Theory | doi=10.1142/2215 | publisher=World Scientific | date=April 1997 | isbn=9810216025}} * {{cite journal|first=Mike |last=Shulman |title=Not every pseudoalgebra is equivalent to a strict one |journal=[[Advances in Mathematics]] |volume=229 |number=3 |year=2012 |pages=2024–2041 |arxiv=1005.1520 |doi=10.1016/j.aim.2011.01.010 |doi-access=free}} * {{cite journal | first1=Mikhail M. | last1=Kapranov | authorlink1=Mikhail Kapranov | title=The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation | journal=Journal of Pure and Applied Algebra | volume=85 | issue=2 | date=1993 | pages=119–142 | doi=10.1016/0022-4049(93)90049-Y | doi-access=}} * {{cite journal | last1=Reiner | first1=Victor | last2=Ziegler | first2=Günter M. | authorlink2=Günter M. Ziegler | title=Coxeter-associahedra | journal=[[Mathematika]] | volume=41 | issue=2 | date=1994 | pages=364–393 | doi=10.1112/S0025579300007452}} == External links == *https://ncatlab.org/nlab/show/homotopy+coherent+diagram *https://unapologetic.wordpress.com/2007/07/01/the-strictification-theorem/ {{topology-stub}} [[Category:Homotopy theory]]'