Category Theory in Context

CATEGORY

THEORY

IN

CONTEXT

Emily Riehl

DOVER PUBLICATIONS, INC., Mineola, New York

Chapter 6 is adapted with permission from Chapter 1 of Categorical Homotopy Theory by Emily Riehl, Cambridge University Press.

Copyright © 2014 by Emily Riehl

Copyright

Copyright © 2016 by Emily Riehl

All rights reserved.

Bibliographical Note

Category Theory in Context is a new work, first published by Dover Publications, Inc., in 2016.

International Standard Book Number

ISBN-13: 978-0-486-80903-8

ISBN-10: 0-486-80903-X

Manufactured in the United States by RR Donnelley

80903X01 2016

www.doverpublications.com

To Peter Johnstone, whose beautiful Part III lectures provided my first acquaintance with category theory and form the skeleton of this book.

and

To Martin Hyland, who guided my initial explorations of this subject’s frontiers and inspired my aspirations to think categorically.

The aim of theory really is, to a great extent, that of systematically organizing past experience in such a way that the next generation, our students and their students and so on, will be able to absorb the essential aspects in as painless a way as possible, and this is the only way in which you can go on cumulatively building up any kind of scientific activity without eventually coming to a dead end.

M.F. Atiyah, How research is carried out [Ati74]

Contents

Preface

Sample corollaries

A tour of basic categorical notions

Note to the reader

Notational conventions

Acknowledgments

Chapter 1.Categories, Functors, Natural Transformations

1.1.Abstract and concrete categories

1.2.Duality

1.3.Functoriality

1.4.Naturality

1.5.Equivalence of categories

1.6.The art of the diagram chase

1.7.The 2-category of categories

Chapter 2.Universal Properties, Representability, and the Yoneda Lemma

2.1.Representable functors

2.2.The Yoneda lemma

2.3.Universal properties and universal elements

2.4.The category of elements

Chapter 3.Limits and Colimits

3.1.Limits and colimits as universal cones

3.2.Limits in the category of sets

3.3.Preservation, reflection, and creation of limits and colimits

3.4.The representable nature of limits and colimits

3.5.Complete and cocomplete categories

3.6.Functoriality of limits and colimits

3.7.Size matters

3.8.Interactions between limits and colimits

Chapter 4.Adjunctions

4.1.Adjoint functors

4.2.The unit and counit as universal arrows

4.3.Contravariant and multivariable adjoint functors

4.4.The calculus of adjunctions

4.5.Adjunctions, limits, and colimits

4.6.Existence of adjoint functors

Chapter 5. Monads and their Algebras

5.1.Monads from adjunctions

5.2.Adjunctions from monads

5.3.Monadic functors

5.4.Canonical presentations via free algebras

5.5.Recognizing categories of algebras

5.6.Limits and colimits in categories of algebras

Chapter 6.All Concepts are Kan Extensions

6.1.Kan extensions

6.2.A formula for Kan extensions

6.3.Pointwise Kan extensions

6.4.Derived functors as Kan extensions

6.5.All concepts

Epilogue: Theorems in Category Theory

E.1.Theorems in basic category theory

E.2.Coherence for symmetric monoidal categories

E.3.The universal property of the unit interval

E.4.A characterization of Grothendieck toposes

E.5.Embeddings of abelian categories

Bibliography

Catalog of Categories

Glossary of Notation

Index

PREFACE

Atiyah described mathematics as the science of analogy. In this vein, the purview of category theory is mathematical analogy. Category theory provides a cross-disciplinary language for mathematics designed to delineate general phenomena, which enables the transfer of ideas from one area of study to another. The category-theoretic perspective can function as a simplifying¹ abstraction, isolating propositions that hold for formal reasons from those whose proofs require techniques particular to a given mathematical discipline.²

A subtle shift in perspective enables mathematical content to be described in language that is relatively indifferent to the variety of objects being considered. Rather than characterize the objects directly, the categorical approach emphasizes the transformations between objects of the same general type. A fundamental lemma in category theory implies that any mathematical object can be characterized by its universal property—loosely by a representation of the morphisms to or from other objects of a similar form. For example, tensor products, free constructions, and localizations are characterized by universal properties in appropriate categories, or mathematical contexts. A universal property typically expresses one of the mathematical roles played by the object in question. For instance, one universal property associated to the unit interval identifies self-homeomorphisms of this space with re-parameterizations of paths. Another highlights the operation of gluing two intervals end to end to obtain a new interval, the construction used to define composition of paths.

Certain classes of universal properties define blueprints which specify how a new object may be built out of a collection of existing ones. A great variety of mathematical constructions fit into this paradigm: products, kernels, completions, free products, gluing constructions, and quotients are all special cases of the general category-theoretic notion of limits or colimits, a characterization that makes it easy to define transformations to or from the objects so-defined. The input data for these constructions are commutative diagrams, which are themselves a vehicle for mathematical definitions, e.g., of rings or algebras, representations of a group, or chain complexes.

Important technical differences between particular varieties of mathematical objects can be described by the distinctive properties of their categories: that rings have all limits and colimits while fields have few, that a continuous bijection defines an isomorphism of compact Hausdorff spaces but not of generic topological spaces. Constructions that convert mathematical objects of one type into objects of another type often define transformations between categories, called functors. Many of the basic objects of study in modern algebraic topology and algebraic geometry involve functors and would be impossible to define without category-theoretic language.

Category theory also contributes new proof techniques, such as diagram chasing or arguments by duality; Steenrod called these methods abstract nonsense.³ The aim of this text is to introduce the language, philosophy, and basic theorems of category theory. A complementary objective is to put this theory into practice: studying functoriality in algebraic topology, naturality in group theory, and universal properties in algebra.

Practitioners often assert that the hard part of category theory is to state the correct definitions. Once these are established and the categorical style of argument is sufficiently internalized, proving the theorems tends to be relatively easy.⁴ Indeed, the proofs of several propositions appearing in this text are left as exercises, with confidence that the reader will eventually find it more efficient to supply their own arguments than to read the author’s.⁵ The relative simplicity of the proofs of major theorems occasionally leads detractors to assert that there are no theorems in category theory. This is not at all the case! Counterexamples abound in the text that follows. A short list of further significant theorems, beyond the scope of a first course but not too far to be out of the reach of comprehension, appears as an epilogue.

Sample corollaries

It is difficult to preview the main theorems in category theory before developing fluency in the language needed to state them. (A reader possessing such fluency might wish to glance ahead to §E.1.) Instead, here are a few corollaries, results in other areas of mathematics that follow trivially as special cases of general categorical results that are proven in this text.

As an application of the theory of equivalence between categories:

COROLLARY 1.5.13. In a path-connected space, any choice of basepoint yields an isomorphic fundamental group.

A fundamental lemma in category theory has the following two results as corollaries:

COROLLARY 2.2.9. Every row operation on matrices with n rows is defined by left multiplication by some n × n matrix, namely the matrix obtained by performing the row operation on the identity matrix.

COROLLARY 2.2.10. Any group is isomorphic to a subgroup of a permutation group.

A special case of a general result involving the interchange of limits and colimits is:

COROLLARY 3.8.4. For any pair of sets X and Y and any function f : X × Y →

whenever these infima and suprema exist.

The following five results illustrate a few of the many corollaries of a common theorem, which describes one consequence of a type of duality enjoyed by certain pairs of mathematical constructions:

COROLLARY 4.5.4. For any function f : A → B, the inverse image function f−1 : PB → PA between the power sets of A and B preserves both unions and intersections, while the direct image function f∗ : PA → PB only preserves unions.

COROLLARY 4.5.5. For any vector spaces U, V, W,

COROLLARY 4.5.6. For any cardinals α, β, γ, cardinal arithmetic satisfies the laws:

COROLLARY 4.5.7. The free group on the set X Y is the free product of the free groups on the sets X and Y.

COROLLARY 4.5.8. For any R–S bimodule M, the tensor product M ⊗S − is right exact.

Finally, a general theorem that recognizes categories whose objects bear some sort of algebraic structure has a number of consequences, including:

COROLLARY 5.6.2. Any bijective continuous function between compact Hausdorff spaces is a homeomorphism.

This is not to say that category theory necessarily provides a more efficient proof of these results. In many cases, the proof that general consensus designates the most elegant reflects the categorical argument. The point is that the category-theoretic perspective allows for an efficient packaging of general arguments that can be used over and over again and eliminates contextual details that can safely be ignored. For instance, our proof that the tensor product commutes with the direct sum of vector spaces will not make use of any bases, but appeals instead to the universal properties of the tensor product and direct sum constructions.

A tour of basic categorical notions

… the science of mathematics exemplifies the interdependence of its parts.

Saunders Mac Lane, Topology and logic as a source of algebra [ML76]

A category is a context for the study of a particular class of mathematical objects. Importantly, a category is not simply a type signature, it has both nouns and verbs, containing specified collections of objects and transformations, called morphisms,⁶ between them. Groups, modules, topological spaces, measure spaces, ordinals, and so forth form categories, but these classifications are not the main point. Rather, the action of packaging each variety of objects into a category shifts one’s perspective from the particularities of each mathematical sub-discipline to potential commonalities between them. A basic observation along these lines is that there is a single categorical definition of isomorphism that specializes to define isomorphisms of groups, homeomorphisms of spaces, order isomorphisms of posets, and even isomorphisms between categories (see Definition 1.1.9).

Mathematics is full of constructions that translate mathematical objects of one kind into objects of another kind. A construction that converts the objects in one category into objects in another category is functorial if it can be extended to a mapping on morphisms in such a way that composites and identity morphisms are preserved. Such constructions define morphisms between categories, called functors. Functoriality is often a key property: for instance, the chain rule from multivariable calculus expresses the functoriality of the derivative (see Example 1.3.2(x)). In contrast with earlier numerical invariants in topology, functorial invariants (the fundamental group, homology) tend both to be more easily computable and also provide more precise information. While the Euler characteristic can distinguish between the closed unit disk and its boundary circle, an easy proof by contradiction involving the functoriality of their fundamental groups proves that any continuous endomorphism of the disk must have a fixed point (see Theorem 1.3.3).

On occasion, functoriality is achieved by categorifying an existing mathematical construction. Categorification refers to the process of turning sets into categories by adding morphisms, whose introduction typically demands a re-interpretation of the elements of the sets as related mathematical objects. A celebrated knot invariant called the Jones polynomial must vanish for any knot diagram that presents the unknot, but its categorification, a functor⁷ called Khovanov homology, detects the unknot in the sense that any knot diagram whose Khovanov homology vanishes must represent the unknot. Khovanov homology converts an oriented link diagram into a chain complex whose graded Euler characteristic is the Jones polynomial.

A functor may describe an equivalence of categories, in which case the objects in one category can be translated into and reconstructed from the objects of another. For instance, there is an equivalence between the category of finite-dimensional vector spaces and linear maps and a category whose objects are natural numbers and whose morphisms are matrices (see Corollary 1.5.11). This process of conversion from college linear algebra to high school linear algebra defines an equivalence of categories; eigenvalues and eigenvectors can be developed for matrices or for linear transformations, it makes no difference.

Treating categories as mathematical objects in and of themselves, a basic observation is that the process of formally turning around all the arrows in a category produces another category. In particular, any theorem proven for all categories also applies to these opposite categories; the re-interpretation of the result in the opposite of an opposite category yields the statement of the dual theorem. Categorical constructions also admit duals: for instance, in Zermelo–Fraenkel set theory, a function f : X → Y is defined via its graph, a subset of X × Y isomorphic to X. The dual presentation represents a function via its cograph, a Y-indexed partition of X Y. Categorically-proven properties of the graph representation will dualize to describe properties of the cograph representation.

Categories and functors were introduced by Eilenberg and Mac Lane with the goal of giving precise meaning to the colloquial usage of natural to describe families of isomorphisms. For example, for any triple of -vector spaces U, V, W, there is an isomorphism

between the set of linear maps U ⊗ V → W and the set of linear maps from U to the vector space Hom(V, W) of linear maps from V to W. This isomorphism is natural in all three variables, meaning it defines an isomorphism not simply between these sets of maps but between appropriate set-valued functors of U, V, and W. Chapter 1 introduces the basic language of category theory, defining categories, functors, natural transformations, and introducing the principle of duality, equivalences of categories, and the method of proof by diagram chasing.

In fact, the isomorphism (0.0.1) defines the vector space U ⊗ V by declaring that linear maps U ⊗ V → W correspond to linear maps U → Hom(V, W), i.e., to bilinear maps U × V → W. This definition is sufficiently robust that important properties of the tensor product—for instance its symmetry and associativity—can be proven without reference to any particular construction (see Proposition 2.3.9 and Exercise 2.3.ii). The advantages of this approach compound as the mathematical objects so-described become more complicated.

In Chapter 2, we study such definitions abstractly. A characterization of the morphisms either to or from a fixed object describes its universal property; the cases of to or from are dual. By the Yoneda lemma—which, despite its innocuous statement, is arguably the most important result in category theory—every object is characterized by either of its universal properties. For example, the Sierpinski space is characterized as a topological space by the property that continuous functions X → S correspond naturally to open subsets of X. The complete graph on n vertices is characterized by the property that graph homomorphisms G → Kn correspond to n-colorings of the vertices of the graph G with the property that adjacent vertices are assigned distinct colors. The polynomial ring [x1,…, xn] is characterized as a commutative unital ring by the property that ring homomorphisms [x1,…, xn] → R correspond to n-tuples of elements (r1,…, rn) ∈ R. Modern algebraic geometry begins from the observation that a commutative ring can be identified with the functor that it represents.

The idea of probing a fixed object using morphisms abutting to it from other objects in the category gives rise to a notion of generalized elements (see Remark 3.4.15). The elements of a set A are in bijection with functions ∗ → A with domain a singleton set; a generalized element of A is a morphism X → A with generic domain. In the category of directed graphs, a parallel pair of graph homomorphisms ϕ, ψ : A B can be distinguished by considering generalized elements of A whose domain is the free-living vertex or the free-living directed edge.⁸ A related idea leads to the representation of a topological space via its singular complex.

The Yoneda lemma implies that a general mathematical object can be represented as a functor valued in the category of sets. A related classical antecedent is a result that comforted those who were troubled by the abstract definition of a group: namely that any group is isomorphic to a subgroup of a permutation group (see Corollary 2.2.10). A deep consequence of these functorial representations is that proofs that general categorically-described constructions are isomorphic reduce to the construction of a bijection between their set-theoretical analogs (for instance, see the proof of Theorem 3.4.12).

Chapter 3 studies a special case of definitions by universal properties, which come in two dual forms, referred to as limits and colimits. For example, aggregating the data of the cyclic p-groups / pn and homomorphisms between them, one can build more complicated abelian groups. Limit constructions build new objects in a category by imposing equations on existing ones. For instance, the diagram of quotient homomorphisms

has a limit, namely the group p of p-adic integers: its elements can be understood as tuples of elements (an ∈ / pn)n∈ that are compatible modulo congruence. There is a categorical explanation for the fact that p is a commutative ring and not merely an abelian group: each of these quotient maps is a ring homomorphism, and so this diagram and also its limit lifts to the category of rings.⁹

By contrast, colimit constructions build new objects by gluing together existing ones. The colimit of the sequence of inclusions

is the Prüfer p-group , an abelian group which can be presented via generators and relations as

The inclusion maps are not ring homomorphisms (failing to preserve the multiplicative identity) and indeed it turns out that the Prüfer p-group does not admit any non-trivial multiplicative structure.

Limits and colimits are accompanied by universal properties that generalize familiar universal properties in analysis. A poset (A, ≤) may be regarded as a category whose objects are the elements a ∈ A and in which a morphism a → a′ is present if and only if a ≤ a′. The supremum of a collection of elements {ai}i∈I, an example of a colimit in the category (A, ≤), has a universal property: namely to prove that

is equivalent to proving that ai ≤ a for all i ∈ I. The universal property of a generic colimit is a generalization of this, where the collection of morphisms (ai → a)i∈I is regarded as data, called a cone under the diagram, rather than simply a family of conditions. Limits have a dual universal property that specializes to the universal property of the infimum of a collection of elements in a poset.

Chapter 4 studies a generalization of the notion of equivalence of categories, in which a pair of categories are connected by a pair of opposite-pointing translation functors called an adjunction. An adjunction expresses a kind of duality between a pair of functors, first recognized in the case of the construction of the tensor product and hom functors for abelian groups (see Example 4.3.11). Any adjunction restricts to define an equivalence between certain subcategories, but categories connected by adjunctions need not be equivalent. For instance, there is an adjunction connecting the poset of subsets of n and the poset of subsets of the ring [x1,…, xn] that restricts to define an equivalence between Zariski closed subsets and radical ideals (see Example 4.3.2). Another adjunction encodes a duality between the constructions of the suspension and of the loop space of a based topological space (see Example 4.3.14).

When a forgetful functor admits an adjoint, that adjoint defines a free (or, less commonly, the dual cofree) construction. Such functors define universal solutions to optimization problems, e.g., of adjoining a multiplicative unit to a non-unital ring. The existence of free groups or free rings have implications for the constructions of limits in these categories (namely, Theorem 4.5.2); the dual properties for colimits do not hold because there are no cofree groups or rings in general. A category-theoretic re-interpretation of the construction of the Stone–Čech compactification of a topological space defines a left adjoint to any limit-preserving functor between any pair of categories with similar set-theoretic properties (see Theorem 4.6.10 and Example 4.6.12).

Many familiar varieties of algebraic objects—such as groups, rings, modules, pointed sets, or sets acted on by a group—admit a free–forgetful adjunction with the category of sets. A special property of these adjoint functors explains many of the common features of the categories of algebras that are presented in this manner. Chapter 5 introduces the categorical approach to universal algebra, which distinguishes the categories of rings, compact Hausdorff spaces, and lattices from the set-theoretically similar categories of fields, generic topological spaces, and posets. The former categories, but not the latter, are categories of algebras over the category of sets.

The notion of algebra is given a precise meaning in relation to a monad, an endofunctor that provides a syntactic encoding of algebraic structure that may be borne by objects in the category on which it acts. Monads are also used to construct categories whose morphisms are partially-defined or non-deterministic functions, such as Markov kernels (see Example 5.2.10), and are separately of interest in computer science. A key result in categorical universal algebra is a vast generalization of the notion of a presentation of a group via generators and relations, such as in (0.0.2), which demonstrates that an algebra of any variety can be presented canonically as a coequalizer¹⁰ of a pair of maps from a free algebra on the relations to a free algebra on the generators.

The concluding Chapter 6 introduces a general formalism that can be used to redefine all of the basic categorical notions introduced in the first part of the text. Special cases of Kan extensions define representable functors, limits, colimits, adjoint functors, and monads, and their study leads to a generalization of, as well as a dualization of, the Yoneda lemma. In the most important cases, a Kan extension can be computed by a particular formula, which specializes to give the construction of a representation for a group induced from a representation for a subgroup (see Example 6.2.8), to provide a new way to think about the collection of ultrafilters on a set (see Example 6.5.12), and to define an equivalence of categories connecting sheaves on a space with étale spaces over that space (see Exercise 6.5.iii).

A brief detour introduces derived functors, which are certain special Kan extensions that are of great importance in homological algebra and algebraic topology. A recent categorical discovery reveals that a common mechanism for constructing point-set level derived functors yields total derived functors with superior universal properties (see Propositions 6.4.12 and 6.4.13). A final motivation for the study of Kan extensions reaches beyond the scope of this book. The calculus of Kan extensions facilitates the extension of basic category theory to enriched, internal, fibered, or higher-dimensional contexts, which provide natural homes for more sophisticated varieties of mathematical objects whose transformations have some sort of higher-dimensional structure.

Note to the reader

The text that follows is littered with examples drawn from a broad range of mathematical areas. The examples are included for color or historical context but are never essential for understanding the abstract category theory. In principle, one could study category theory immediately after learning some basic set theory and logic, as no other prerequisites are strictly required, but without some level of mathematical maturity it would be difficult to see what the point of it all is. We hope that the majority of examples are comprehensible in outline, even if the details are unfamiliar, but if this is not the case, it is not worth stressing over. Inevitably, given the diversity of mathematical tastes and experiences, the examples presented here will seldom be optimized for any particular individual, and indeed, each reader is encouraged to search for their own contexts in which to explore categorical ideas.

Notational conventions

An arrow symbol →, either in a display or in text, is only ever used to denote a morphism in an appropriate category. In particular, the objects surrounding it necessarily lie in a common category. Double arrows ⇒ are reserved for natural transformations, the notation used to suggest the intuition that these are some variety of 2-dimensional morphisms. The symbol , read as maps to, appears occasionally when defining a function between sets by specifying its action on particular elements. The symbol is used in a less technical sense to mean something along the lines of yields or leads to or can be used to construct. If the presence of certain morphisms implies the existence of another morphism, the latter is often depicted with a dashed arrow to suggest the correct order of inference.¹¹

We use as an abbreviation for a parallel pair of morphisms, i.e., for a pair of morphisms with common source and target, and as an abbreviation for an opposing pair of morphisms with sources and targets swapped.

Italics are used occasionally for emphasis and to highlight technical terms. Boldface signals that a technical term is being defined by its surrounding text.

The symbol = is reserved for genuine equality (with := used for definitional equality), with ≅ used instead for isomorphism in the appropriate ambient category, by far the more common occurrence.

Acknowledgments

Many of the theorems appearing here are standard fare for a first course on category theory, but the examples are not. Rather than rely solely on my own generative capacity, I consulted a great many people while preparing this text and am grateful for their generosity in sharing ideas.

To begin, I would like to thank the following people who responded to a call for examples on the n-Category Café and the categories mailing list: John Baez, Martin Brandenburg, Ronnie Brown, Tyler Bryson, Tim Campion, Yemon Choi, Adrian Clough, Samuel Dean, Josh Drum, David Ellerman, Tom Ellis, Richard Garner, Sameer Gupta, Gejza Jenča, Mark Johnson, Anders Kock, Tom LaGatta, Paul Levy, Fred Linton, Aaron Mazel-Gee, Jesse McKeown, Kimmo Rosenthal, Mike Shulman, Peter Smith, Arnaud Spiwack, Ross Street, John Terilla, Todd Trimble, Mozibur Ullah, Enrico Vitale, David White, Graham White, and Qiaochu Yuan.

In particular, Anders Kock suggested a more general formulation of the chain rule expresses the functoriality of the derivative than appears in Example 1.3.2(x). The expression of the fundamental theorem of Galois theory as an isomorphism of categories that appears as Example 1.3.15 is a favorite exercise of Peter May’s. Charles Blair suggested Exercise 1.3.iii and a number of expository improvements to the first chapter. I learned about the unnatural isomorphism of Proposition 1.4.4 from Mitya Boyarchenko. Peter Haine suggested Example 1.4.6, expressing the Riesz representation theorem as a natural isomorphism of Banach space-valued functors; Example 3.6.2, constructing the path-components functor, and Example 3.8.6. He also contributed Exercises 1.2.v, 1.5.v, 1.6.iv, 3.1.xii, and 4.5.v and served as my consultant.

John Baez reminded me that the groupoid of finite sets is a categorification of the natural numbers, providing a suitable framework in which to prove certain basic equations in elementary arithmetic; see Example 1.4.9 and Corollary 4.5.6. Juan Climent Vidal suggested using the axiom of regularity to define the non-trivial part of the equivalence of categories presented in Example 1.5.6 and contributed the equivalence of plane geometries that appears as Exercise 1.5.viii. Samuel Dean suggested Corollary 1.5.13, Fred Linton suggested Corollary 2.2.9, and Ronnie Brown suggested Example 3.5.8. Ralf Meyer suggested Example 2.1.5(vi), Example 5.2.6(ii), Corollary 4.5.8, and a number of exercises including 2.1.ii and 2.4.v, which he used when teaching a similar course. He also pointed me toward a simpler proof of Proposition 6.4.12.

Mozibur Ullah suggested Exercise 3.5.vii. I learned of the non-natural objectwise isomorphism appearing in Example 3.6.5 from Martin Brandenburg who acquired it from Tom Leinster. Martin also suggested the description of the real exponential function as a Kan extension appearing in Example 6.2.7. Andrew Putman pointed out that Lang’s Algebra constructs the free group on a set using the construction of the General Adjoint Functor Theorem, recorded as Example 4.6.6. Paul Levy suggested using affine spaces to motivate the category of algebras over a monad, as discussed in Section 5.2; a similar example was suggested by Enrico Vitale. Dominic Verity suggested something like Exercise 5.5.vii. Vladimir Sotirov pointed out that the appropriate size hypotheses were missing from the original statement of Theorem 6.3.7 and directed me toward a more elegant proof of Lemma 4.6.5.

Marina Lehner, while writing her undergraduate senior thesis under my direction, showed me that an entirely satisfactory account of Kan extensions can be given without the calculus of ends and coends. I have enjoyed, and this book has been enriched by, several years of impromptu categorical conversations with Omar Antolín Camarena. I am extremely appreciative of the careful readings undertaken by Tobias Barthel, Martin Brandenburg, Benjamin Diamond, Darij Grinberg, Peter Haine, Ralf Meyer, Peter Smith, and Juan Climent Vidal, who each sent detailed lists of corrections and suggestions. I would also like to thank John Grafton and Janet Kopito, the acquisitions and in-house editors at Dover Publications, and David Gargaro for his meticulous copyediting.

I am grateful for the perspicacious comments and questions from those who attended the first iteration of this course at Harvard—Paul Bamberg, Nathan Gupta, Peter Haine, Andrew Liu, Wyatt Mackey, Nat Mayer, Selorm Ohene, Jacob Seidman, Aaron Slipper, Alma Steingart, Nithin Tumma, Jeffrey Yan, Liang Zhang, and Michael Fountaine, who served as the course assistant—and the second iteration at Johns Hopkins—Benjamin Diamond, Nathaniel Filardo, Alex Grounds, Hanveen Koh, Alex Rozenshteyn, Xiyuan Wang, Shengpei Yan, and Zhaoning Yang. I would also like to thank the Departments of Mathematics at both institutions for giving me opportunities to teach this course and my colleagues there, who have created two extremely pleasant working environments. While these notes were being revised, I received financial support from the National Science Foundation Division of Mathematical Sciences DMS-1509016.

My enthusiasm for mathematical writing and patience for editing were inherited from Peter May, my PhD supervisor. I appreciate the indulgence of my collaborators, my friends, and my family while this manuscript was in its final stages of preparation. This book is dedicated to Peter Johnstone and Martin Hyland, whose tutelage I was fortunate to come under in a pivotal year spent at Cambridge supported by the Winston Churchill Foundation of the United States.

¹In his mathematical notebooks, Hilbert formulated a 24th problem (inspired by his work on syzygies) to develop a criterion of simplicity for evaluating competing proofs of the same result [TW02].

²For example, the standard properties of induced representations (Frobenius reciprocity, transitivity of induction, even the explicit formula) are true of any construction defined as a left Kan extension; character tables, however, are non-formal.

³Lang’s Algebra [Lan02, p. 759] supports the general consensus that this was not intended as an epithet:

In the forties and fifties (mostly in the works of Cartan, Eilenberg, MacLane, and Steenrod, see [CE56]), it was realized that there was a systematic way of developing certain relations of linear algebra, depending only on fairly general constructions which were mostly arrow-theoretic, and were affectionately called abstract nonsense by Steenrod.

⁴A famous exercise in Lang’s Algebra asks the reader to Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book [Lan84, p. 175]. Homological algebra is the subject whose development induced Eilenberg and Mac Lane to introduce the general notions of category, functor, and natural transformation.

⁵In the first iteration of the course that inspired the writing of these lecture notes, the proofs of several major theorems were also initially left to the exercises, with a type-written version appearing only after the problem set was due.

⁶The term morphism is derived from homomorphism, the name given in algebra to a structure-preserving function. Synonyms include arrow (because of the notation →) and map (adopting the standard mathematical colloquialism).

⁷Morally, one could argue that functoriality is the main innovation in this construction, but making this functoriality precise is somewhat subtle [CMW09].

⁸The incidence relation in the graph A can be recovered by also considering the homomorphisms between these graphs.

⁹The lifting of the limit is considerably more subtle than the lifting of the diagram. Results of this nature motivate Chapter 5.

¹⁰A coequalizer is a generalization of a cokernel to contexts that may lack a zero homomorphism.

¹¹Readers who dislike this convention can simply connect the dots.

CATEGORY

THEORY

IN

CONTEXT

CHAPTER 1

Categories, Functors, Natural Transformations

Frequently in modern mathematics there occur phenomena of naturality.

Samuel Eilenberg and Saunders Mac Lane, Natural isomorphisms in group theory [EM42b]

A group extension of an abelian group H by an abelian group G consists of a group E together with an inclusion of G E as a normal subgroup and a surjective homomorphism E H that displays H as the quotient group E/G. This data is typically displayed in a diagram of group homomorphisms:¹

A pair of group extensions E and E′ of G and H are considered to be equivalent whenever there is an isomorphism E ≅ E′ that commutes with the inclusions of G and quotient maps to H, in a sense that is made precise in §1.6. The set of equivalence classes of abelian group extensions E of H by G defines an abelian group Ext(H, G).

In 1941, Saunders Mac Lane gave a lecture at the University of Michigan in which he computed for a prime p that Ext( [ ]/ , ) ≅ p, the group of p-adic integers, where [ ]/ is the Prüfer p-group. When he explained this result to Samuel Eilenberg, who had missed the lecture, Eilenberg recognized the calculation as the homology of the 3-sphere complement of the p-adic solenoid, a space formed as the infinite intersection of a sequence of solid tori, each wound around p times