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Category Theory 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 .
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