A SHORT INTRODUCTION TO CATEGORY THEORY September 4, 2019 Contents 1. Category theory 1 1.1. Definition of a category 1 1.2. Natural transformations 3 1.3. Epimorphisms and monomorphisms 5 1.4. Yoneda Lemma 6 1.5. Limits and colimits 7 1.6. Free objects 9 References 9 1. Category theory 1.1. Definition of a category. Definition 1.1.1. A category C is the following data (1) a class ob(C), whose elements are called objects; (2) for any pair of objects A; B of C a set HomC(A; B), called set of morphisms; (3) for any three objects A; B; C of C a function HomC(A; B) × HomC(B; C) −! HomC(A; C) (f; g) −! g ◦ f; called composition of morphisms; which satisfy the following axioms (i) the sets of morphisms are all disjoints, so any morphism f determines his domain and his target; (ii) the composition is associative; (iii) for any object A of C there exists a morphism idA 2 Hom(A; A), called identity morphism, such that for any object B of C and any morphisms f 2 Hom(A; B) and g 2 Hom(C; A) we have f ◦ idA = f and idA ◦ g = g. Remark 1.1.2. The above definition is the definition of what is called a locally small category. For a general category we should admit that the set of morphisms is not a set. If the class of object is in fact a set then the category is called small. For a general category we should admit that morphisms form a class. We do not really need this definition even if it is useful in general. We remark that in the above definition, for any object A, the identity morphism is unique. Definition 1.1.3. A morphism f : A −! B is invertible if there exists a morphism g : B −! A, called inverse, such that g ◦ f = idA and f ◦ g = idB. In such a case we will say that A is isomorphic to B and f is also called an isomorphism. Even if ob(C) is not in general a set we will use sometimes the notation A 2 ob(C) to say that A is an object of C. Exercise 1.1.4. Prove that the inverse of an invertible morphism is unique. We observe that given a category C we can construct a category Co which has the same objects of C and such that for any two objects A; B we set HomCo (A; B) = HomC(B; A). So essentially we reverse all the arrows Examples 1.1.5. Here is some examples of categories. (1) Given a set X we can consider the category X whose objects are the elements of X and any morphism is the identity. If X is empty we obtain the empty category. (2) Set whose objects are sets and morphisms are functions. 1 2 A SHORT INTRODUCTION TO CATEGORY THEORY (3) Grp whose objects are groups and morphisms are group homomorphisms. (4) Ab whose objects are abelian groups and morphisms are group homomorphisms (5) Rng whose objects are rings with unity and morphisms are ring homomorphisms preserving unity. (6) CRng whose objects are commutative rings with unity and morphisms are ring homomorphisms preserving unity. (7) R-Mod whose objects are left R-modules and morphisms are left R-module homomorphisms. if R is a field k we call this category Vectk. We call FVectk the category of finite dimensional vector spaces. (8) Mod-R whose objects are right R-modules and morphisms are right R-module homomorphisms. (9) Top whose objects are topological spaces and morphisms are continuous functions. (10) Top∗ whose objects are topological spaces with a selected base point and morphisms are contin- uous functions which preserve base points. (11) Every pre-order set, i.e. every pair (X; ≤) with X a non-empty set and ≤ a transitive and reflexive relation on X, gives arise to a category whose objects are the elements of x and for any pair of elements x and y one defines Hom(x; y) as follows : it has an element (x; y) if x ≤ y and it is empty otherwise. The composition is given by (y; z) ◦ (x; y) := (x; z). The opposite category is the category which arises by (X; ≥). The preorder is a partial order, i.e. is anti-symmetric, if and only if the identities are the only isomorphisms. (12) A group G can be considered as a category BG with one object and where the morphisms are given by the elements of the group and the composition is given by the multiplication. Clearly any morphism has a two-sided inverse under composition. Definition 1.1.6. Given two categories C and D, a functor F : C −! D is a map that assigns to each object A of C an object F (A) of D and to each morphism f : A −! B of C a morphism F (f): F (A) −! F (B) such that F (f ◦ g) = F (f) ◦ F (g) and F (idA) = idA. We remark that one can sometimes speak about covariant functor and contravariant functor. In fact a covariant functor from C to D is just a functor as defined before, while a contravariant functor from C to D is a functor from Co to D and it is often denoted F : C −! D specifying that it is contravariant. So a contravariant functor just reverses arrows. Examples 1.1.7. (1) For any category C we have the functor identity idC defined in the obvious way. (2) For any category C and any set X to give a functor X !C is equivalent to give a set of objects fAxgx2X of C parametrized by X. (3) For any category C and any group G to give a functor BG !C is equivalent to give an object A of C with a group morphism G ! AutC(A), where AutC(A) is the set of isomorphisms of A in A, called automorphisms of A, which is clearly a group. We say that G acts over A. So if, for instance, C = Sets to give a functor BG !C is equivalent to give a G-set A. If H is another group then to give a functor BG ! BH is equivalent to give a group morphism G ! H since H = AutBH(?). A (4) For any object of a category C we can construct the covariant functor h := HomC(A; −) from C 0 to Sets which associates to any object B the set HomC(A; B) and to any morphism f : B ! B the function 0 f∗ : HomC(A; B) −! HomC(A; B ) g 7−! f ◦ g Similarly we can construct the contravariant functor hA := HomC(−;A) from C to Sets which 0 associates to any object B the set HomC(B; A) and to any morphism f : B ! B in C the function ∗ 0 f : HomC(B ;A) −! HomC(B; A) g 7−! g ◦ f (5) We have a natural functor F or : Grp −! Sets called forgetful. This functor forgets the group structure, so to any group it associates its underline set and to any group homomorphism it associates the same homomorphism seen as set function. In general a forgetful functor is a functor which simply forget some additional structure or property. Of course there are many other examples of forgetful functors. For instance F or : Rng −! Grp, F or : R-Mod −! Ab or F or : Ab −! Grp. (6) We have the functor Ab : Grp −! Ab which associates to any group G his abelianized Gab. Prove that it is a functor. A SHORT INTRODUCTION TO CATEGORY THEORY 3 (7) The topological fundamental group gives a functor Π1 : Top∗ −! Grp. (8) Taking the dual of a vector space over a field k gives a contravariant functor k-Mod −! k-Mod. (9) For any commutative ring with unity R we can consider the group GLn(R) which consists of invertible matrices with coefficients in R. It is easy to verify that in fact this gives a functor GLn : Crng −! Grp. If n = 1 this functor associates to any ring R the group of invertible elements R∗. In this case we denote it by (−)∗. (10) The construction of the dual of a vector space gives arise to a contravariant functor (−)_ : Vectk −! Vectk. (11) For any object X of a category D and for any category C we can construct the constant functor X which associates to any object of C the object X and to any morphism the identity morphism idX . Remark that if we have two functors F : C −! D and G : D −! E we can construct the composition G ◦ F . In fact if A is an object of C we define (G ◦ F )(A) := G(F (A)) and, if f : A −! B is a morphism of C, we define G ◦ F (f) := G(F (f)). Exercise 1.1.8. Prove that the above composition of functors is well defined and prove that it verifies the associative property. Remark 1.1.9. So we can consider the category Cat of small categories, where morphisms are the functors. If we consider locally small categories then we obtain a large category. Definition 1.1.10. A functor F : C −! D is (i) an isomorphism of categories if there exists a functor G : D −! C such that F ◦ G = idD and G ◦ F = idC . Any such a G is called inverse of F ; (ii) full if, for any pair of objects A and B of C, the function FA;B : HomC(A; B) −! HomD(F (A);F (B)) is surjective; (iii) faithful if FA;B is injective for any pair of objects A; B of C.