WOMP 2017: Category Theory

Minh-Tam Trinh

1. Categories

1.1.A category C consists of: (1) A class of objects. (2) For any objects x and y, a class HomC.y; x/ of morphisms from y to x. (3) For any objects x; y; z, a composition law

(1.1) HomC.y; x/ HomC.z; y/ HomC.z; x/ ı W  ! that satisfies: (a) Identity. For all objects x, there is a morphism idx x x such that f idx f W ! ı D and idx g g. ı D (b) Associativity. We have f .g h/ .f g/ h. ı ı D ı ı We will write C0, resp. C1, for the class of objects, resp. morphisms, of C.

Example 1.2. The categories Set, Grp, Ab, Ring, R-Mod, and k-Vect are self-explanatory.

Example 1.3. For each of

(1.2) topological, C k, C , C !;  D 1 there is a category -Man where the objects are -manifolds and the morphisms are maps   preserving the -structure. For instance, if topological, then the morphisms are the   D continuous maps. (In this case, we denote the category just by Man.)

Example 1.4. If P is a poset, then there is a category in which the objects are the elements of P and we add a morphism y x if and only if y x. Note that the existence of idx ! Ä corresponds to the reflexive property x x. Ä

Example 1.5. We can view any group G as a category G in its own right: Take G0 to consist of a single object , and take G1 G, each element g G being viewed as a morphism  D 2 g . The identity and associativity axioms for G correspond precisely to the identity W  !  and associativity axioms for G.

Example 1.6. Consider the category FSet in which the objects are finite sets and the morphisms are arbitrary maps between sets. Is FSet0 a set or a proper class? Category Theory 2

1.7. The last example suggests that even though C0 is usually freakishly large, we don’t actually conceptualize it that way, and don’t need to conceptualize it that way. To this end, we say that a morphism f y x is an isomorphism if and only if there is another W ! morphism g x y such that W !

(1.3) f g idx and g f idy : ı D ı D An automorphism of x is an isomorphism from x to itself. The first unspoken paradigm of category theory is: Isomorphic objects x0 and x00 can be treated as the same object x (once we interpret the morphisms from x0 to x00 as endomorphisms of x). Objects in FSet are isomorphic if and only if they have the same cardinality, so to study FSet, it suffices to choose one object of each cardinality and study the morphisms between these. This is the 5000-year-old1 principle of mathematics known as “counting.” 1.8. As our examples suggest, most categories in a “working mathematician’s” life appear in one or more of the following uses.

(1) Linear-algebraic. The prototype is Ab. Categories C where HomC.y; x/ is equipped with the structure of an abelian group for all x and y, and is additive in a natural ı sense. Keywords are additive category, abelian category, category of complexes, derived category, etc. (2) Tensor-algebraic. The prototype is C-Vect. Categories C in which C0 is equipped with an operation behaving like the composition law in a monoid, and especially ˝ like a tensor product. Often overlaps with (1). Keywords are monoidal category, symmetric monoidal, braided monoidal, rigid monoidal, etc. (3) Geometric-topological. The prototype is Man. Categories of spaces, or “generalized spaces” (e.g., topoi), defined either by chart/atlas conditions or by sheaf data (see below). (4) Order-theoretic. Categories built from posets, or possibly graphs (in which case they are called quivers), appearing in heterogeneous ways in combinatorics, representation theory, and analysis.

2. Functors 2.1. It is natural to propose a notion of “morphism of categories.” If C; D are categories, then a functor F D C consists of maps D0 C0 and D1 C1 that respect W ! ! ! composition of morphisms, i.e.,

(1) F.idx/ id . D F .x/ (2) F .g f / F .g/ F .f /. ı D ı A bare map D0 C0 is functorial iff it can be extended to a functor D C. We write Cat ! ! for the category in which the objects are (other) categories and the morphisms are functors. Example 2.2. The map that sends a manifold to its ith singular homology is functorial for all i, as a continuous map M N induces a morphism Hi .M / Hi .N /. ! ! Example 2.3. If Z is the poset of integers under and P is an arbitrary poset, then a functor Ä Z P is equivalent to an ascending chain in P . ! 1Older? Maybe it depends on your interpretation of the Ishango Bone. Category Theory 3

Example 2.4. Let G be a (discrete) group. Recall that we can view it as a category G. Check that a complex representation of G is the same as a functor G C-Vect. ! Example 2.5. A groupoid is a category in which every morphism is an isomorphism. For any manifold M , let the fundamental groupoid of M be the category …1.M / in which the objects are the points of M and the morphisms are oriented paths between points. Then …1 induces a functor Man Cat. ! 2.6. Henceforth we assume our categories are always locally small, meaning for each x and y, the class HomC.y; x/ is a set and not a proper class. We say that a functor F D C is: W ! (1) Faithful iff F .f / F .g/ implies f g for all morphisms f; g. That is, F is D D injective on morphisms. (2) Full iff every morphism from F .y/ to F .x/ takes the form F .f / for some morphism F y x. That is, F is surjective on morphisms. W ! (3) Fully faithful iff F is both full and faithful.

If F is faithful, then we say that D is a subcategory of C. Each subclass of C0 gives rise to a full subcategory of C, consisting of the objects in that subclass and all morphisms in C between them. We say that F is essentially surjective iff every object of C is isomorphic to one of the form F .x/ for some object x of D. We say that F is an equivalence of categories iff it is both fully faithful and essentially surjective. We can now augment the first paradigm with a second one: Equivalent categories C0 and C00 can be treated as the same category C. Example 2.7. If k < `, then the forgetful functor C `-Man C k-Man is faithful but not 1 ! full. For instance, not every C function on R is C 1. Example 2.8. Our earlier observation about FSet can be restated as: FSet is equivalent to its full subcategory of objects of the form 1; : : : ; n . f g 2.9. Natural transformations are to functors as functors are to categories. Explicitly, if F;G are functors D C, then a natural transformation or 2-functor ˆ F G consists ! W ! of a choice of morphism ˆ.x/ F .x/ G.x/ in C for each object x of D, such that W ! (2.1) G.f / ˆ.y/ ˆ.x/ F .f / ı D ı for all morphisms f y x. This condition says that ˆ transforms F .f / into G.f /. It is W ! equivalent to saying that the two possible compositions in the diagram

ˆ.y/ F .y/ G.y/

(2.2) F .f / G.f /

ˆ.x/ F .x/ G.x/ are equal. One says that the diagram commutes. Much of category theory can be easily expressed and remembered in the language of commutative diagrams. Category Theory 4

Example 2.10. There is an analogy

functors continuous maps natural transformations homotopies(2.3) W WW W that we can make precise using the functor …1. Indeed, if f; g M N are continuous W ! maps, then they induce functors …1.f /; …1.g/ …1.M / …1.N /, and a homotopy W ! f g induces a natural transformation …1.f / …1.g/. ! ! Example 2.11. Let G be a discrete group and G the corresponding category. Just as a representation of G is a functor G C-Vect, a morphism of representations is a natural ! transformation between such functors.

3. Universal Properties

3.1. Using a superlative adjective is basically the same as stating that an object or morphism satisfies a universal property: largest abelian quotient of a group least upper bound of a set of numbers coarsest topology making a map continuous We will not actually give explicit definitions in this section. Instead, we will give a litany of examples and exhort the reader to supply the definitions on her own.

Example 3.2. Possibly the most fundamental example: (1) The empty set is initial in the category Set, in the sense that it admits a unique map to every other set. (2) Up to isomorphism, the one-element set is final in Set, in the sense that every other set admits a unique map to a one-element set.

Example 3.3. If N G, then the map G G=N is initial among all group homomor- C ! phisms out of G that annihilate N , in the sense that if

 (3.1) G H ! annihilates N , then there is a unique map  G=N H such that the composition 0 W !  (3.2) G G=N 0 H ! ! equals . We say that  factors through G G=N uniquely. ! In the case where N ŒG; G, the subgroup of G generated by all commutators, Gab D D G=ŒG; G is initial among all homomorphisms from G into an abelian group. That is, Gab is the largest abelian quotient of G.

3.4. In the real world, a thing that is superlative in some property might not be unique (the brightest of days, the reddest of tomatoes). In mathematics, an object that satisfies a universal property need not be unique, but it will nonetheless be unique up to a unique isomorphism. Category Theory 5

3.5. Limits and colimits are a particular case of universal properties. In order to explain them, we introduce cones and cocones. If C is a category and D a subcategory of C, then a cone over D consists of an object x of C and morphisms from x to each object of D, commuting with all the morphisms internal to D.A cocone is defined similarly, except the morphisms go in reverse, viz., from each object of D to x.

Example 3.6. If D looks like

b

(3.3) a d

c

(identity morphisms and compositions omitted), then a cone over D looks like the collection of dashed arrows in the diagram:

b

(3.4) x a d

c provided that the dashed morphisms commute with the arrows in D.

3.7. The limit, or pro-limit (projective limit), of D is a cone final among all cones of D (if it exists). Similarly, the colimit, or ind-limit (inductive limit), of D is a cocone initial among all cocones of D (if it exists). Sometimes these are denoted using symbols like lim D and colim D.

Example 3.8. Let R be the poset of real numbers under . If S is a subset of R, viewed as Ä a subcategory, then: (1) The limit of S is the greatest lower bound of S. (2) The colimit of S is the least upper bound of S. This example shows concretely that neither limits nor colimits need exist.

Example 3.9. Fix a set X and consider the category in which the objects are subsets of X and the morphisms are inclusion maps. Fix a set of objects Ui , and let C be the full subcategory they generate. T (1) The limit of C is the intersection i Ui . It is final among all subsets that, for every i, are contained in Ui . S (2) The colimit of C is the union i Ui . It is initial among all subsets that, for every i, contain Ui . Category Theory 6

Example 3.10. Suppose we have topological spaces X1;X2;Y and maps fi Y Xi . W ! These spaces and maps generate a subcategory of the category of topological spaces. Its colimit is the quotient space formed by gluing together X1 and X2 via the relations f1.y/ ` f D f2.y/ y Y , sometimes denoted X1 Y X2. g 2 3.11. We say that a limit, resp. colimit, is directed or filtered iff its defining subcategory is formed from a lower, resp. upper, semilattice. More specifically, it is sequential iff the semilattice is isomorphic to N under or the reverse poset. Ä 1 Example 3.12. In the category of (abelian) groups, the filtered colimit of the groups n Z 1 1 under the inclusion maps Z Z for d n is the group Q. d  n j Example 3.13. The power-series ring ZŒŒt is equipped with a quotient map

n n (3.5) n ZŒŒt ZŒŒt=t ZŒt=t W ! ' n m for all n. If m n, then m factors through n, via the map m;n ZŒt=t ZŒt=t . Ä W ! It turns out that ZŒŒt is the sequential limit, in the category of rings, of the subcategory n formed by the rings ZŒt=t and the maps m;n. 3.14. Products and coproducts are, respectively, the limits and colimits where the defining subcategory does not involve any morphisms. In other words, they are defined solely in terms of a subclass or collection of objects.

Example 3.15. If Xi is a collection of topological spaces, then: f Qg (1) Their product i Xi is the product of their underlying sets, equipped with the coarsest topology making the projection to each Xi continuous. It is final among all spaces Y equipped with a continuous map Y Xi for each i. ` ! (2) Their coproduct i Xi is the disjoint union of their underlying sets, equipped with the finest topology making the inclusion of each Xi continuous. It is initial among all spaces Y equipped with a continuous map Xi Y for each i. ` ` ! Note that X1 X2 is precisely X1 X2 in the notation of Example 3.10. ; Example 3.16. If R is a commutative ring with unity and Mi i is a collection of R-modules, f g then: Q (1) Their product i Mi can be represented by the module consisting of arbitrary se- quences .mi /i with mi Mi . 2 L (2) Their coproduct, conventionally denoted i Mi , can be represented by the module consisting of sequences .mi /i such that mi 0 for all but finitely many i. D Note that when the collection is finite, coproduct and product are isomorphic. This property is one of the axioms used to define additive categories, which behave like R-Mod.

Example 3.17. Keep R as before. If Ri i is a collection of R-algebras, then: f g (1) Their product, as R-algebras, has the same underlying R-module as their product as R-modules. N (2) Their coproduct, as R-algebras, is the tensor product i Ri . Category Theory 7

In particular, in the category of arbitrary commutative unital rings, the coproduct of rings A and B is A B. ˝Z Remark 3.18. Warning—by contrast, tensor products of R-modules are usually not the coproducts of their constituents!

Example 3.19. In our fundamental Example 3.10, the empty set is the product over the empty collection of sets, whereas the one-element set is the coproduct over the empty collection.

4. Adjunctions 4.1. Suppose we are given categories C; D and functors F D C and G C D. We W ! W ! say that .F; G/ form an adjoint pair, or an adjunction, iff there are bijections

(4.1) HomC.F .y/; x/ HomD.y; G.x// ' for every object x of C and y of D, which are functorial in the sense that for all morphisms x x and y y, the induced diagrams 0 ! 0 !

HomC.F .y/; x0/ HomD.y; G.x0// (4.2)

HomC.F .y/; x/ HomD.y; G.x// and

HomC.F .y0/; x/ HomD.y0; G.x// (4.3)

HomC.F .y/; x/ HomD.y; G.x// commute. (People rarely write out the verification of this commutativity condition in practice.) In this case, we say that F , resp. G, is the left adjoint, resp. right adjoint, in the pair. This section will first enumerate several examples, then explain how adjunctions interact with limits and colimits (in Proposition 4.9.

Example 4.2. The inspiration for the concept itself is the so-called currying adjunction. If A; B; C are sets, then there is a functorial bijection:

(4.4) HomSet.A B;C/ HomSet.A; HomSet.B; C //:  ' Here, F is the endofunctor of Set that sends A A B and G is the endofunctor of Set 7!  that sends C HomSet.B; C /. 7! Example 4.3. The previous example can be upgraded to a version for topological spaces, provided the spaces are sufficiently nice. For example, this works for the full subcategory CG of “compactly-generated spaces.” If Y and Z are such spaces, then we can turn Category Theory 8

HomCG.Y; Z/ into an object of CG by endowing it with the compact-open topology. For any spaces X;Y;Z in CG, we have

(4.5) HomCG.X Y;Z/ HomCG.X; HomCG.Y; Z//  ' just as before.

Example 4.4. In the category of R-modules for a commutative unital ring R, the analogue of the previous two examples is the tensor-hom adjunction of Atiyah–Macdonald fame:

(4.6) HomR.M N;M/ HomR.M ; HomR.N; M // 0 ˝ ' 0 for all R-modules M;M 0;N . There is also a version for modules over noncommutative rings, once the adjectives “left” and “right” are inserted appropriately.

Example 4.5. Let C be the category where the objects are (base)pointed topological spaces and the morphisms are homotopy classes of basepoint-preserving continuous maps. In this category, it is conventional to write ŒY; X to mean HomC.Y; X/. If .X; x/ is an object of C, then we can form its (pointed) suspension

X Œ 1; 1 (4.7) †.X; x/  C ' X 1 x Œ 1; 1  f˙ g [ f g  C and its loopspace

1 (4.8) .X; x/ HomC..S ; 1/; .X; x//: ' One can check that the operations † and  define endofunctors of C for which .†; / is an adjunction.

Example 4.6. If G is a group and H is a subgroup, then there is a restriction functor from representations of G to representations of H, denoted

(4.9) ResG Rep Rep : H W G ! H If G is a Lie group and H is a Lie subgroup, then there are two functors in the reverse direction, called induction and compact induction:

(4.10) IndG Rep Rep and CptIndG Rep Rep : H W H ! G H W H ! G I’m not going to define them here, but it is easy to find the definitions on the Internet. The following are adjunctions:

G G G G (4.11) .CptIndH ; ResH / and .ResH ; IndH /:

Moreover, if G=H is a compact manifold, then CptIndG IndG . In this case, restriction H D H and induction are each both left and right adjoints of each other. This situation is called a Frobenius reciprocity; we say that the (appropriately-defined) group algebra of G is a Frobenius algebra over the group algebra of H. Category Theory 9

4.7. In what follows, the phrase “F commutes with limits” means

“F.lim D/ lim F.D/ for all D”(4.12) D in the obvious sense, and similarly with colimits.

Lemma 4.8. For a fixed object x of a category C, the functor y HomC.x; y/ commutes with limits. 7!

Proposition 4.9. If F is a left, resp. right adjoint, then F commutes with colimits, resp. limits.

Proof. First, prove the Yoneda lemma (see Exercise 5.8). Then consider, e.g., that

HomC.y; G.limi xi // HomC.F .y/; limi xi / D (4.13) limi HomC.F .y/; xi / D limi HomC.y; G.xi // D in the right-adjoint case. Remark 4.10. For the left-adjoint case, one has to introduce the notion of a contravariant functor, or equivalently, of the opposite category of C (just reverse the directions of the morphisms), and establish the correct analogue of Lemma 4.8. 4.11. This result is innocuous but versatile. It immediately tells you, for instance, that tensor products commute with colimits of modules and loopspace operations commute up to homotopy with limits of pointed spaces. I, for one, remember using it a lot in the first-year curriculum.

5. Exercises 5.1. We defined what it meant for a functor to be (essentially) injective on morphisms, surjective on morphisms, and surjective on objects. Why didn’t we define what it meant to be injective on objects?

5.2. Let Mann be the category of n-dimensional (Hausdorff second-countable) compact topological manifolds under continuous maps. For n 0; 1; 2, show that Mann is equivalent D to an (explicit) full subcategory on countably many objects. 5.3. Verify that Examples 3.8-3.13 actually do give limits and/or colimits. 5.4. Verify that Examples 3.15-3.19 actually do give products and/or coproducts. 5.5. Work out the implications of Remark 4.10 and finish the proof of Proposition 4.9. 5.6.A 2-category consists of objects, morphisms, and 2-morphisms, the last of these being “morphisms between morphisms.” Construct a working definition of a 2-category. In what sense(s) can 2-morphisms be composed? 5.7. Look up the definitions of presheaf and sheaf. Find two qualitatively different examples of a presheaf that is not a sheaf. Category Theory 10

5.8. Look up the Yoneda lemma and make sense of it. Relate its meaning to the proverb, “A person becomes a person through other people.”

Good luck in your studies at the University of Chicago!