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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 ∈ C, a set of morphisms HomC (A, B). An element f ∈ 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 ∈ C a distinguished identity morphism 1A , which acts as a twosided identity for composition of morphisms. I.e. for all f ∈ HomC (A, B), one has f ◦ 1A = f , and for all g ∈ HomC (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 C op denote the opposite category of C, which is defined as follows. (1) One has obC = obC op . (2) For all A, B objects in C op (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 morphisms in C op , we define g ◦ f (the composition in C op ) as f ◦C g (the composition in C). Some other important definitions which were covered in class are summarised in the following 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 ∈ 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 ∈ C. 1 2. Functors Let C and D be two categories. A (covariant) Functor between C and D, denoted by F F : C → D or C − → D, consists of the following data: (1) (2) (3) (4) For each A ∈ C, an object F (A) ∈ D. For each morphism f : A → B in C, a morphism F (f ) : F (A) → F (B) in D. F is compatible with compositions: F (f ◦C g) = F (f ) ◦D F (g). F is compatible with identities: F (1A ) = 1F (A) for all A ∈ C. A contravariant functor between C and D is just a covariant functor F : C op → 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 ∈ 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): F (A) F (f ) η(A) F (B) / / η(B) G(A) G(f ) G(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 ∈ 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 ∈ C, we have a natural functor HomC (A, •) : C → Set defined as follows: (1) For B ∈ C, the object of Set associated to B is the set HomC (A, B). (2) For a morphism f : B → B 0 in C, the map HomC (A, B) → HomC (A, B 0 ) 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 ∈ C, the object of Set associated to B is the set HomC (B, A). 2 (2) For a morphism f : B → B 0 in C, the map HomC (B 0 , 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 ∈ 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 ∈ 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 ∈ imF (i.e. A = F (A0 ) and B = F (B0 ) for some A0 , B0 ∈ C), we defined HomimF (A, B) = F (HomC (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