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A Class of Objects Obc, Usually Denoted by Just C BASIC DEFINITIONS IN CATEGORY THEORY MATH 250B ADAM TOPAZ 1. Categories A category C consists of the following data: (1) A class of objects obC, usually denoted by just C. (2) For each A; B 2 C, a set of morphisms HomC(A; B). An element f 2 HomC(A; B) is called a morphism between A and B, and will sometimes be denoted by f : A ! B f or A −! B. (3) An associative composition rule for morphisms f : A ! B and g : B ! C. I.e. this is a map HomC(B; C) × HomC(A; B) ! HomC(A; C) denoted by (f; g) 7! f ◦ g, such that ◦ is associated (when defined). (4) For each object A 2 C a distinguished identity morphism 1A, which acts as a two- sided identity for composition of morphisms. I.e. for all f 2 HomC(A; B), one has C f ◦ 1A = f, and for all g 2 Hom (B; A) one has 1A ◦ g = g. We went over several examples of categories in class, and one can find many other examples all over the place. A trivial, but important construction, is the opposite of a category C, defined as follows. For a category C, we let Cop denote the opposite category of C, which is defined as follows. (1) One has obC = obCop. (2) For all A; B objects in Cop (equiv. objects in C), one has op HomC (A; B) = HomC(B; A): (3) Composition is defined in the natural way: if f : A ! B and g : B ! C are op op morphisms in C , we define g ◦ f (the composition in C ) as f ◦C g (the composition in C). Some other important definitions which were covered in class are summarised in the fol- lowing list: (1) A category C is small if obC is actually a set. (2) A category C is a subcategory of the category D if one has obC ⊂ obD and HomC(A; B) ⊂ HomD(A; B) for all A; B 2 C ⊂ D. (3) In the context of (2) above, we say that C is a full subcategory of D if one further has HomC(A; B) ⊂ HomD(A; B) for all A; B 2 C. 1 2. Functors Let C and D be two categories. A (covariant) Functor between C and D, denoted by F : C!D or C −!DF , consists of the following data: (1) For each A 2 C, an object F (A) 2 D. (2) For each morphism f : A ! B in C, a morphism F (f): F (A) ! F (B) in D. (3) F is compatible with compositions: F (f ◦C g) = F (f) ◦D F (g). (4) F is compatible with identities: F (1A) = 1F (A) for all A 2 C. A contravariant functor between C and D is just a covariant functor F : Cop !D. If F; G : C!D are two functors, then a natural transformation between F and G, η denoted by η : F ! G or F −! G, is defined by the following data: (1) For each A 2 C, a moprhism η(A): F (A) ! G(A). (2) If f : A ! B is a moprhism in C, then one has a commutative diagram (in D): η(A) F (A) / G(A) F (f) G(f) F (B) / G(B) η(B) With this definition, we can consider Cat the category of (small) categories, whose objects are small categories, and whose morphisms are covariant functors. Note that if C; D 2 Cat, then the hom-set HomCat(C; D) can also be given the structure of a category, where morphisms are natural transformations. Some further definitions related to functors are summarized in the following list: (1) A functor F : C!D is called injective if F (A) = F (B) implies A = B. (2) A functor F : C!D is called full resp. faithful if the map F : HomC(A; B) ! HomD(F (A);F (B)) is surjective resp. injective. (3)A fully faithful functor is a functor which is both full and faithful. 3. Representable Functors Let Set denote the category of sets (i.e. objects are sets and morphisms are functions between sets). For a category C and an object A 2 C, we have a natural functor HomC(A; •): C! Set defined as follows: (1) For B 2 C, the object of Set associated to B is the set HomC(A; B). (2) For a morphism f : B ! B0 in C, the map HomC(A; B) ! HomC(A; B0) is simply composition with f, i.e. g 7! f ◦ g. Similarly, we obtain a contravariant functor HomC(•;A): C! Set defined as follows: (1) For B 2 C, the object of Set associated to B is the set HomC(B; A). 2 (2) For a morphism f : B ! B0 in C, the map HomC(B0;A) ! HomC(B; A) is simply pre-composition with f, i.e. g 7! g ◦ f. A covariant functor F : C! Set said to be representable by A 2 C if one has an isomorphism of functors: F ∼= HomC(A; •): Similarly, a contravariant functor G : C! Set is said to be representable by A 2 C if one has an isomorphism of functors: G ∼= HomC(•;A): It is more-or-less standard to call a contravariant functor C! Set a presheaf. This terminology comes from algebraic geometry and modern algebraic topology. We will use this terminology in class. 4. The Image of a Functor Let F : C!D be a functor. The image of F , denoted by imF is a subcategory of D defined as follows: (1) The objects of imF is the sub-class F (obC) of obD. (2) For A; B 2 imF (i.e. A = F (A0) and B = F (B0) for some A0;B0 2 C), we defined imF C Hom (A; B) = F (Hom (A0;B0)): It is easy to see that imF is a subcategory of D. Moreover, imF is a full subcategory if and only if F is full. 5. Isomorphisms and Equivalences of Categories Let C be a category. An isomoprhism f : A ! B in C is just an invertible morphism. I.e f is an isomoprhism if there exists g : B ! A such that f ◦ g = 1B and g ◦ f = 1A. Following the idea that functors are morphisms between categories, we say that a functor F : C!D is an isomoprhism of categories if F is invertible as a functor. I.e. if there exists a functor G : D!C such that G ◦ F = 1C and F ◦ G = 1D. A more subtle notion than that of isomorphism is the notion of an equivalence of categories. We say that a functor F : C!D is an equivalence of categories if there exists a functor ∼ ∼ ∼ G : D!C such that F ◦ G = 1D and G ◦ F = 1C, where = denotes isomorphism of functors (in the category Fun(C; C) resp. Fun(D; D)). The following is more-or-less obvious: (1) If F : C!D is an injective functor which is faithful, then F induces an isomorphism of categories F : C! imF . (2) If F is an injective, fully faithful functor, then imF is a full subcategory of D, which is isomorphic to C. (3) We say that F is essentially surjective if every object of D is isomorphic to some object in imF . It is easy to see that an equivalence of categories is fully faithful and essentially surjective. (4) Conversely, if F is fully faithful and essentially surjective, then F is an equivalence of categories (exercise). 3 We conclude this note witha brief remark which should aid in intuition: The notion of equivalence of categories is analogous to the notion of homotopy equivalence in algebraic topology. The analogy goes like this: if we think of a category as a topological space, then a functor should be considered as a continuous map between two spaces. Thus an isomorphism of categories should be thought of as a homeomorphism of topological spaces. An isomorphism between two functors should then be thought of as a homotopy equivalence between two continuous maps. Two equivalent categories should therefore be thought of as two homotopy-equivalent spaces. This analogy can be made very precise, but the details are beyond the scope of this course. 4.
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