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Brief Introduction to Categories Brief Introduction to Categories Torsten Wedhorn TU Darmstadt Contents 1 Categories 1 2 Functors 3 3 Limits and Colimits 5 4 Special cases of limits and colimits 8 1 Categories Definition 1.1. A category C consists ofy (a) a class Ob(C) of objects, (b) for any two objects X and Y a set C(X; Y ) := HomC(X; Y ) = Hom(X; Y ) of morphisms from X to Y , (c) for any three objects X, Y , Z a composition map C(X; Y ) × C(Y; Z) !C(X; Z); (f; g) 7! g ◦ f such that all morphisms sets are disjoint (hence every morphism f 2 C(X; Y ) has unique source X and a unique target Y ) and such that (1) the composition of morphisms is associative (i.e., for all objects X, Y , Z, W and for all f 2 C(X; Y ), g 2 C(Y; Z), h 2 C(Z; W ) one has h ◦ (g ◦ f) = (h ◦ g) ◦ f, (2) For all objects X there exists a morphism idX 2 C(X; X), called identity of X such that for all objects Y and for all morphisms f 2 C(X; Y ) and g 2 C(Y; X) one has f ◦ idX = f and idX ◦g = g. A category is called finite if it has only finitely many objects and morphisms. Remark 1.2. For every object X of a category C the morphism idX is uniquely deter- ~ ~ ~ mined by Condition (2) for if idX and idX are identities of X, then idX = idX ◦ idX = idX . yHere we ignore all set-theoretical issues. To avoid set-theoretical difficulties one should work with a fixed universe in the sense of [KaSh] Definition 1.1.1 and assume that the sets of morphisms between two objects always lies in the given universe. Then a category C is called small if Ob(C) is in that universe. Moreover it then will be sometimes necessary to pass to a bigger universe, for instance when considering the category of functors between two categories (see Definition 2.4 below). Finally one adds also the axiom that every set is an element of some universe to the axioms of Zermelo-Fraenkel set theory. We refer to [SGA4] Exp. I for details. Alternatively one can also work with classes as explained in [Sch]. 1 Definition and Remark 1.3. A morphism f : X ! Y in a category is called an isomorphism if there exists a morphism g : Y ! X such that f ◦g = idY and g◦f = idX . ∼ We often write f : X ! Y to indicate that f is an isomorphism. We also write X =∼ Y and say that X and Y are isomorphic if there exists an isomorphism X !∼ Y . An morphism (resp. an isomorphism) with the same source and target X is called an endomorphism (resp. an automorphism) of X. Composition yields the structure of a monoid on the set EndC(X) of endomorphisms of X and the structure of a group on the set AutC(X) of automorphisms. Definition 1.4. A subcategory of a category C is a category C0 such that every object of C0 is an object of C and such that C0(X0;Y 0) ⊆ C(X0;Y 0) for any pair (X0;Y 0) of objects of C0, compatibly with composition of morphisms and identity elements. The subcategory C0 is called full if C0(X0;Y 0) = C(X0;Y 0) for all objects X0 and Y 0 of C0. Example 1.5. (1) (Sets) the category of sets: Objects are sets, for two sets X and Y a morphism is a map X ! Y , composition in the category is the usual composition of maps, the identity of a set X is the usual identity map idX . An isomorphism in (Sets) is simply a bijective map. (2) (Grp) the category of groups: Objects are groups, morphisms are group homo- morphisms, composition is the usual composition of group homomorphisms, the identity is the usual identity. An isomorphism in (Grp) is a group isomorphism. (3) (Ab) is the full subcategory of (Grp) of abelian groups: Objects are abelian groups, morphisms are group homomorphisms, composition is the usual composition of group homomorphisms, the identity is the usual identity. (4) Fix a group G. Then (G-Sets) denotes the category of G-sets: Objects are sets with a left action by the group G, morphisms are G-equivariant maps. By (Sets-G) we denote the category of sets with a right action by G. (5) (Ring) the category of rings: Objects are rings (always assumed to have a unit), morphisms are ring homomorphisms (preserving the unit). (6) (Top) the category of topological spaces: Objects are topological spaces, morphisms are continuous maps. An isomorphism is a homeomorphism. (7) Let R be a ring. Then (R-Mod) denotes the category of left R-modules: Objects are left R-modules, morphisms are R-linear maps. An isomorphism is bijective R-linear map because its inverse is automatically R-linear again. If R = k is a field, we obtain the category of k-vector spaces, usually denoted by (k-Vec) instead of (k-Mod). Example 1.6. Let I be a set. (1) A relation ≤ on I is called partial preorder or simply preorder, if i ≤ i for all i 2 I and i ≤ j, j ≤ k imply i ≤ k for all i; j; k 2 I. (2) A preorder ≤ is called partial order or simply order if i ≤ j and j ≤ i imply i = j for all i; j 2 I. (3) A preorder ≤ is called filtered if for all i; j 2 I there exists a k 2 I with i ≤ k and j ≤ k. (4) A partial order ≤ is called total order if for all i; j 2 I one has i ≤ j or j ≤ i. 2 Every preordered set I can be made into a category, again denoted by I, whose objects are the elements of I and for two elements i; j 2 I the set of morphisms I(i; j) consists of one element if i ≤ j and is empty otherwise. There is a unique way to define a composition law in this category. Definition 1.7. Let C be a category. A morphism f : X ! Y in C is called monomor- phism (resp. epimorphism) if for all objects Z in C and all morphisms g; h: Z ! X (resp. g; h: Y ! Z) one has f ◦ g = f ◦ h ) g = h (resp. g ◦ f = h ◦ f ) g = h). Example 1.8. In the category of sets (resp. of groups), monomorphisms are injective maps (resp. group homomorphisms) and epimorphisms are surjective maps (resp. group homomorphisms). In the category of rings the inclusion Z ! Q is a monomorphism and an epimorphism. This gives in particular an example of a morphisms that is a monomorphism and an epimorphism but not an isomorphism. Definition 1.9. For every category C the opposite category, denoted by Copp, is the category with the same objects as C and where for two objects X and Y of Copp we set Copp(X; Y ) := C(Y; X) with the obvious composition law. Definition 1.10. Let C and D be categories. We define the product category C × D by Ob(C ×D) := Ob(C)×Ob(D) and (C ×D)((X1;Y1); (X2;Y2)) := C(X1;X2)×D(Y1;Y2). Composition of morphisms is defined componentwise. 2 Functors Definition 2.1. Given categories C and D, a (covariant) functor F : C!D is given by attaching to each object C of C an object F (C) of D, and to each morphism f : C ! C0 in C a morphism F (f): F (C) ! F (C0), compatible with composition of morphisms, i.e., F (g◦f) = F (g)◦F (f) whenever the composition is defined, and preserving identity elements, i.e., F (idC ) = idF (C). If F : C!D and G: D!E are functors, we write G ◦ F : C!E for the composition. A covariant functor Copp !D is also called a contravariant functor from C to D. By a \functor" we will always mean a covariant functor. If we write F : C!D, then we always mean that F is covariant. A contravariant functor F from C to D will always be denoted by F : Copp !D. It attaches to each object C in C an object F (C) in D and to each morphism f : C ! C0 in C a morphism F (f): F (C0) ! F (C) such that F (g ◦ f) = F (f) ◦ F (g) and F (idC ) = idF (C). Remark 2.2. Let F : C!D be a functor. Let C and C0 be objects in C and let f : C ! C0 be an isomorphism in C. Then F (f) is an isomorphism. 0 Indeed, as f is an isomorphism, there exists a morphism g : C ! C such that g◦f = idC and f ◦ g = idC0 . Hence F (g) ◦ F (f) = F (g ◦ f) = F (idC ) = idF (C) and similarly F (f) ◦ F (g) = idF (C0). Therefore F (f) is an isomorphism. 3 Example 2.3. A simple example is the functor which \forgets" some structure. An example is the functor F : (Grp) ! (Sets) that attaches to every group the underlying set and that sends every group homomorphism to itself (but now considered as a map of sets), Definition and Remark 2.4. For two functors F; G: C!D we call a family of morphisms α(S): F (S) ! G(S) for every object S of C functorial in S if for every morphism f : T ! S in C the diagram α(T ) F (T ) / G(T ) F (f) G(f) α(S) F (S) / G(S) commutes.
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