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Home , Category (mathematics), Concrete category, Epimorphism, Free object, Functor, Inverse limit

A SHORT INTRODUCTION TO THEORY

September 4, 2019

Contents 1. 1 1.1. Definition of a category 1 1.2. Natural transformations 3 1.3. and 5 1.4. 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 ob(C), whose elements are called objects; (2) for any pair of objects A, B of C a HomC(A, B), called set of ; (3) for any three objects A, B, C of C a

HomC(A, B) × HomC(B,C) −→ HomC(A, C) (f, g) −→ g ◦ f, called composition of morphisms; which satisfy the following (i) the sets of morphisms are all disjoints, so any f determines his domain and his target; (ii) the composition is associative; (iii) for any object A of C there exists a morphism idA ∈ Hom(A, A), called morphism, such that for any object B of C and any morphisms f ∈ Hom(A, B) and g ∈ 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 . Even if ob(C) is not in general a set we will use sometimes the notation A ∈ 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 . (4) Ab whose objects are abelian groups and morphisms are group homomorphisms (5) whose objects are rings with unity and morphisms are 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- 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 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- and ≤ a transitive and reflexive 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 is the category which arises by (X, ≥). The is a partial order, i.e. is anti-symmetric, if and only if the identities are the only . (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 F : C −→ D is a 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 {Ax}x∈X 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 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 it associates the same homomorphism seen as set function. In general a is a functor which simply forget some additional structure or property. Of course there are many other examples of forgetful . 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 gives a functor Π1 : Top∗ −→ Grp. (8) Taking the of a vector over a field k gives a contravariant functor k-Mod −→ k-Mod. (9) For any 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 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 . 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. If any such a function is bijective then F is called fully faithful; (iv) essentially surjective if any object of D is isomorphic to an object of type F (A) with A object of C. In fact the notion of isomorphism of category is too restrictive and we will see later the more interesting notion of equivalence of categories. Definition 1.1.11. A S of a category C is category such that ob(S) is a subclass in ob(C) and for any pair A, B of objects of S, HomS (A, B) is a of HomC(A, B). The obvious functor S −→ C is called inclusion. We say that S is a full subcategory of C if the inclusion functor is full. Clearly any inclusion is faithful. Example 1.1.12. (1) The functor BG → C of Example 1.1.7 is faithful if and only if the action is faithful, i.e. G → AutC(A) is injective. (2) Examples of inclusions are given by forgetful functors given in Example 1.1.7(5). Between them only Ab −→ Grp is a full inclusion. Not all forgetful functors are inclusions. For instance the forgetful functor Top∗ −→ Top, which drops the base point, is not an inclusion. Another example of full subcategory is R-Div in R-Mod. (3) We can consider the functor Grp → Cat, given by G 7→ BG. This functor is fully faithful. This follows by Example 1.1.7(3). So this means that any group G is completely determined by BG and so to give a group G is equivalent to give a category with one object such that all morphisms are invertible. Exercise 1.1.13. Let F : C −→ D be a functor and let f : A −→ B be a morphism in C. (1) Prove that if f is an isomorphism then F (f) is an isomorphism. (2) Suppose that F is fully faithfully. Prove that F (f) is an isomorphism if and only if f is an isomorphism. 1.2. Natural transformations. In this subsection we define morphisms between functors. This notion will let us to determine when two functors are equivalent. The notion of equivalence between functors is better than the naive notion of isomorphism. Definition 1.2.1. Let F,G : C −→ D be two functors. A from F to G, noted by α : F ⇒ G, is a collection {α(A)}A∈ob(C) such that 4 A SHORT INTRODUCTION TO CATEGORY THEORY

(i) α(A): F (A) −→ G(A) is a morphism of D and (ii) for any morphism f : A −→ B of C the following diagram

α(A) F (A) / G(A)

F (f) G(f)  α(B)  F (B) / G(B) commutes. Clearly for any functor F there exists the identity natural transformation defined in the obvious manner. Given two natural transformations α : F ⇒ G and β : G ⇒ H between functors F, G, H : C −→ D, the vertical composition β ◦ α is defined as follows: for any object A of C we define (β ◦ α)(A) as β(A) ◦ α(A). It is a natural transformation (see Exercise 1.2.3 below). Remark 1.2.2. This composition allows to define the category (using a slightly more general notion of category respect that one we gave) of functors where morphisms are natural transformations, modulo the Remark 1.1.2. One can also define horizontal composition. We will not do it. Exercise 1.2.3. Prove that the composition of two natural transformations is a natural transformation. And moreover prove that the composition is associative. Definition 1.2.4. A natural transformation α : F ⇒ G between functors F,G : C −→ D is a natural equivalence if there exists a natural transformation β : G ⇒ F such that α ◦ β = idG and β ◦ α = idF . In such a case we note F =∼ G. Exercise 1.2.5. Prove that a natural transformation α : F ⇒ G between functors F,G : C −→ D is a natural equivalence if and only if for any object A of C, the morphism α(A): F (A) −→ G(A) is invertible in D. Definition 1.2.6. A functor F : C −→ D is an equivalence of categories if there exists a functor ∼ ∼ G : D −→ C such that F ◦ G = idD and G ◦ F = idC. The functor G is called the quasi-inverse of F . Exercise 1.2.7. Prove that the inverse of a functor is also a quasi-inverse. And moreover prove that a quasi-inverse, if it exists, is unique.

∨ Examples 1.2.8. (1) If we compose twice the contravariant functor (−) : Vectk −→ Vectk we ∨∨ ∨∨ obtain the bidual (−) which is covariant. We have a natural transformation idVectk ⇒ (−) given as follows : for any vector space V we define the V −→ V ∨∨ by v 7→ (f 7→ f(v)). This construction satisfies the given in the definition of natural transformation. It is well know that if the of the vector space is finite the above linear map is an isomorphism and so the above natural transformation restricted to the category of finite ∨ vector spaces is a natural equivalence. This can be also rephrased saying that (−) : FVectk −→ FVectk is an anti-equivalence (i.e. contravariant equivalence) of categories where the quasi- inverse is itself. It also justifies the usual phrase ” a finite dimensional vector space is canonically isomorphic to its bidual”. This is not true for the dual since to define the isomorphism we need ∨ the choice of a base, and so it is not possible to define a natural equivalence idFVectk ⇒ (−) . ∗ (2) The gives a natural transformation det : GLn −→ (−) (Prove it!). (3) A functor equivalent to one of the functors of Example 1.1.7(2), for some object A, is called representable. Proposition 1.2.9. A functor F : C −→ D is an equivalence of categories if and only if F is fully faithful and essentially surjective. Proof. ⇒). Let G be the quasi-inverse of F . Clearly F is essentially surjective since any object A of C is isomorphic to F (G(A)). We now prove that F is fully faithfully. We first prove a special case.

Lemma 1.2.10. Let H : C −→ C be a functor and α : idC ⇒ H a natural equivalence. Then H is fully faithfully.

Proof. We have to prove that the function HomC(A, B) −→ HomC(H(A),H(B)) is bijective. It is easy to see, using the fact that α is a natural transformation, that HomC(A, B) −→ HomC(H(A),H(B)) is given by f 7−→ α(B) ◦ f ◦ α(A)−1. This is clearly a .  A SHORT INTRODUCTION TO CATEGORY THEORY 5

We come back to the general case. Since we have an equivalence idC ⇒ G ◦ F then by the previous lemma we have that

HomC(A, B) −→ HomC(G ◦ F (A),G ◦ F (B)) is bijective. Since we have the factorization

(1.1) HomC(A, B) −→ HomD(F (A),F (B)) −→ HomC(G ◦ F (A),G ◦ F (B)) then HomC(A, B) → HomD(F (A),F (B)) is injective and HomD(F (A),F (B)) → HomC(G ◦ F (A),G ◦ F (B)) is surjective. In the same way, using the natural equivalence idD ⇒ F ◦ G we have that 0 0 0 0 0 0 0 HomD(A ,B ) → HomC(G(A ),G(B )) is injective for any pair of objects A ,B of D. Taking A = F (A) 0 and B = F (B) we get that HomD(F (A),F (B)) → HomC(G ◦ F (A),G ◦ F (B)) is also injective. So one deduces, by (1.1), that HomC(A, B) → HomC(F (A),F (B)) is finally bijective. ⇐). Using the global of choice 1 and the the fact that F is essentially surjective, for any object D of D we can choose an object G(D) of C and an isomorphism α(D): D −→ F (G(D)). We first prove that G : D −→ C gives a functor. Let g : D → E be a morphism in D. Since F is fully faithful we can −1 define G(g): G(D) → G(E) as the unique morphism in C such that F (G(g)) = αE ◦ g ◦ αD. It is easy to verify that G preserves identity and composition. Now we stress that by definition of G we have that for any morphism g : D −→ E of D the following diagram D / F (G(D)) α(D) g F (G(g))   E / F (G(E)) α(E) commutes. So we have a natural transformation α : idD ⇒ F ◦ G, which is an equivalence, by the Exercise 1.2.5, since α(D) is invertible for any object D. It remains to prove that there exists a natural equivalence β : idC ⇒ G◦F . Let A be an object of C. Let us consider the isomorphism α(F (A)) : F (A) −→ F (G(F (A)). Since F is fully faithfully there exists a unique morphism β(A): A −→ G(F (A)) such that F (β(A)) = α(F (A)). Using again that F is fully faithful one proves (do it!) that β : idC ⇒ G ◦ F is a natural transformation. Moreover, for any A, we have that α(F (A)) is invertible and F (β(A)) = α(F (A)) so by Exercise 1.1.13 we have that β(A) is invertible which implies, by Exercise 1.2.5, that β is a natural equivalence. We have finally proved that G is the quasi-inverse of F . 

1.3. Epimorphisms and monomorphisms. Definition 1.3.1. Let f : A −→ A0 be a morphism in a category C. (1) We will say that f is a if, for any object B of C, the function

0 HomC(B,A) −→ HomC(B,A ) g 7→ f ◦ g

is injective. It will be noted by A,→ A0. (2) We will say that f is a if, for any object B of C, the function

0 HomC(A ,B) −→ HomC(A, B) g 7→ g ◦ f

0 is injective. It will be note by A  A Concretely the above definitions mean the following : f is a monomorphism if f ◦ g = f ◦ g0 implies g = g0, and f is an epimorphism if g ◦ f = g ◦ f 0 implies g = g0 where g and g0 are morphisms with the opportune target and domain. Lemma 1.3.2. Monomorphisms in a category C correspond to epimorphisms in Co. Remark 1.3.3. Since Coo = C we also have that monomorphisms of Co correspond to epimorphisms of C.

1This axiom extends the usual for sets to classes. 6 A SHORT INTRODUCTION TO CATEGORY THEORY

Proof. Let g : A −→ A0 be a morphism in C. Then it gives a morphism go : A0 −→ A in Co. Now the function 0 HomC(B,A) −→ HomC(B,A ) g 7→ f ◦ g is exactly the function 0 HomCo (A, B) −→ HomCo (A ,B) go 7→ go ◦ f. Then the first is injective if and only if the second one is injective. This means that g is a monomorphism o if and only if g is an epimorphism.  The above lemma is useful since, if we have to prove a statement about monomorphisms and epimor- phisms in a general category, it is sufficient to prove it in just one of the two cases. Exercises 1.3.4. (1) Prove that an isomorphism is both a monomorphism and an epimorphism. (2) Prove that in Sets (resp. Ab 2) monomorphisms correspond to injective functions and epimor- phisms to surjective functions. (3) Prove that in Rng the obvious inclusion Z −→ Q is an epimorphism. (4) Prove that in R-Div the morphism Q → Q/Z is a monomorphism. (5) Let Haus be the full subcategory of Top whose objects are Hausdorff topological spaces. Prove that an open and dense inclusion U −→ X is an epimorphism. (6)A is a pair (C,F ) where C is a category and F : C → Sets a faithful functor, which is usually a forgetful functor. Prove that in a concrete category any morphism f with F (f) surjective (resp. injective) is an epimorphism (resp. monomorphism). this to the examples (3) and (4). (7) One can generalize the above example (6). Prove that if F : C → D is a faithful functor then a morphism f : A → B is a monomorphism (resp. epimorphism) if F (f): F (A) → F (B) is a monomorphism (resp. epimorphism). (8) Prove that the composition of monomorphisms (resp. epimorphisms) is a monomorphism. (9) Prove that if g ◦ f is a monomorphism (resp. epimorphism) then f is a monomorphism (resp. g is an epimorphism). By the above exercises we see that an homomorphism which is both monomorpism and epimorphism is not necessarily an isomorphism. 1.4. Yoneda Lemma. We can construct a (possibly large) category SetsC, where objects are functors from C to Sets and morphisms are natural transformations3. Lemma 1.4.1 (Yoneda Lemma). For any category C, functor F : C → Sets and X object of C we have that X ψX,F : HomSetsC (h ,F ) → F (X) X given, for any % ∈ HomSetsC (h ,F ), by ψX,F (%) := %(X)(idX ), is bijective. Moreover this bijection is natural in X and F . Proof. We observe that for any f ∈ hX (Y ), with Y ∈ ob(C), we have that X f = h (f)(idX ). X So we have that %(Y )(f) = %(Y ) ◦ h (f)(idX ). Since % is a natural transformation then the following diagramm commutes %(X) hX (X) / F (X)

hX (f) F (f)  %(Y )  hX (Y ) / F (Y ) we also have X %(Y )(f) = %(Y ) ◦ h (f)(idX ) = F (f)(%(X)(idX )).

So % is uniquely determined by ψX,f (%). So ψX,f is injective. Conversely if a ∈ F (X) then we can define, for any Y ∈ ob(C) and f : X → Y , %(Y )(f) := F (f)(a).

2This is true also in Grp but to prove the statement for epimorphisms is harder. 3The set of natural transformation could not be a set if C is not small. A SHORT INTRODUCTION TO CATEGORY THEORY 7

This gives a natural transformation such that, by construction, ψX,F (%) = %(X)(idX ) = F (idX )(a) = a. It is easy to see that ψX,F is natural in X and F .  Now we can consider the functor h− : Co −→ SetsC X − given by X 7→ h . If we have f : X → Y in C then hX,Y (f) is given by composition. Corollary 1.4.2. The functor h− is fully faithful, i.e. for any X,Y ∈ ob(C) there is a natural isomor- X Y phism HomC(Y,X) ' HomSetsC (h , h ). Proof. In the previous Lemma with F = hY we have a natural isomorphism X Y ψX,F : HomSetsC (h , h ) → HomC(Y,X). − − Indeed it is easy to see that ψX,hY ◦ hX,Y : HomC(Y,X) → HomC(Y,X) is the identity. So hX,Y is an isomorphism for any X,Y ∈ ob(C).  The previous corollary asserts that any (locally small) category can be embedded in a category of functors. We could obtain similar results using the contravariant functor h−. For instance we would have o C a fully faithful functor h− : C → Sets . This essentially means that morphism from (or to) a fixed object determine uniquely the object, a unique isomorphism. Like in Physics. ”You work at a particle accelerator. You want to understand some particle. All you can do are throw other particles at it and see what happens. If you understand how your mystery particle responds to all possible test particles at all possible test energies, then you know everything there is to know about your mystery particle.” (Ravi Vakil) Example 1.4.3. (1) Let X be a set. We can think it as a category X, see Example 1.1.5(1). We remark that X = Xo. So we consider the functor X h− : X −→ Sets . − Since for any a, b ∈ X we have that HomX(a, b) is empty or a we have that h factors X through Sets≤1 where Sets≤1 is the of cardinal at most one. But in fact this category is equivalent to 2 = {∅, ∗}, full subcategory of Sets, as it is easy to see.4 It is also not difficult to see that 2X is equivalent to the category given by (P(X), ⊆) So finally we have that the inclusion X h− : X −→ 2 given by Yoneda Lemma is just the usual inclusion of a set in his . Indeed for any X element a ∈ X the function ha : X → 2 is given by ha(a) = ∗ and ha(b) = ∅ if a 6= b. (2) Let G be a group and let us consider the associated category BG (see Example 1.1.5(12)). We observe that BG = BGo. We saw that that SetsBG is equivalent to the category of G- sets, i.e. sets with a G-action where morphisms are functions G-equivariant. So the functor BG h− : BG −→ Sets associated to the unique object of BG the G-set G with the usual action. So essentially Yoneda Lemma associates to the group G the G-set G. (3) It is easy to see that if n is the category induced by ({1, . . . , n)}, ≤) then the Yoneda factorizes through n → n + 1, given by i 7→ i + 1. 1.5. Limits and colimits. Definition 1.5.1. Let F : I → C be a functor. A colimit for F is an object X of C with the following : there exists a natural transformation γ : F ⇒ X such that any natural transformation α : F ⇒ A, with A object of C, factorizes uniquely by γ. We remark that to give a natural transformation X ⇒ A is equivalent to give a morphism X → A. More generally a natural transformation α : F ⇒ A is nothing else that a collection of morphisms α(i): F (i) −→ A, indexed by objects i of I, such that for any morphism aij : i → j in I, the following diagram F (a ) F (i) ij / F (j)

α(i) α(j) ! } A commutes. The pair (A, α) is called a co- of F .

4This category is also clearly equivalent to the associated category to the order sets ({0, 1}, ≤). 8 A SHORT INTRODUCTION TO CATEGORY THEORY

So, explicitly, the above definition is as follow. The colimit is a co-cone (X, γ) of F such that for any co-cone (A, α) of F there exists a unique morphism f : X −→ A such that the following diagram

F (a ) F (i) ij / F (j)

γ(i) γ(j) ! } α(i) X α(j)

f   Ð A commutes. Exercise 1.5.2. Prove that, if it exists, the colimit is unique up to an unique isomorphism. This is a general fact for objects defined by universal properties. By the above exercise the colimit is unique, if it exists, and it is denoted by lim F . −→ Examples-Exercises 1.5.3. (1) If I is the empty category the colimit is called initial object of C. Prove that an object I of a category C is initial if and only if for any object A of C there exists a unique morphism I → A. (2) Let I be the category given by a set I. In this case the colimit is called which is denoted by qi∈I F (i). The coproduct will be called finite if the set I is finite. Write down explicitly in this case the universal property. Show that if C = Sets the coproduct is the indexed by I, and if C = Ab it is the indexed by I. And show that the direct sum is not the coproduct5 in Grp. (3) Let I be the category with three objects A, B, C of C and only two morphisms f : C −→ A and g : C −→ B of C, in addition to the identities maps. In this case the is called push-out. Of course if C is the initial object this coincide with the coproduct of two objects of the point (ii). Write down explicitly the universal property in this case. (4) Let I be the category with two objects A, B of C and only two morphisms f1, f2 : A −→ B of C, in addition to the identities maps. In this case the coproduct is called co-equalizer. Write down explicitly the universal property in this case. Prove that if C = Ab and f2 = 0 then the is nothing else that the usual Coker f1 equipped by the obvious surjection B → Coker f1. Prove that any co-equalizer is an epimoprhism. (5) Let I be the category given by a preordered set (I, ≤) (see Example 1.1.5(11)). We also suppose that it is directed, i.e. for any i, j ∈ X there exists k ∈ I such that i ≤ k and j ≤ k. In this case the colimit is called and noted by lim A , where A = F (i). Write down explicitly the −→ i i i∈I universal property in this case. Prove that if C = Sets and, for any i ≤ j, F (i) −→ F (j) is the inclusion then the direct limit is the union (not disjoint) indexed by I. One can also construct limits, which are the colimits in the opposite category Co. Essentially in the above constructions all the arrows are reversed. Exercise 1.5.4. Write down explicitly the definition of limit of a functor F . By the uniqueness of colimits follows the uniqueness of limits. And, if it exists, it is noted by lim F . ←− Examples-Exercises 1.5.5. (1) If C is the empty category the limit is called terminal object. Con- cretely an object T of a category C is called terminal if for any object A of C there exists a unique morphism A → T . (2) Let I be the category given by a set I. In this case the limit is called and noted by Q i∈I F (i). The product will be called finite if the set I is finite. Write down explicitly in this case the universal property. Show that if C = Sets, Grp, Ab the product is the usual product indexed by I. (3) Let I be the category with three objects A, B, C of C and only two morphisms f : A −→ C and g : B −→ C of C, in addition to the identities maps. In this case the limit is called pull-back and it is denoted by A ×C B. Of course if C is the terminal object this coincide with the product of two objects of the point (ii). Write down explicitly the universal property in this case. Prove that if C = Sch this is the usual fiber product .

5In Grp the coproduct is given by the which we are not going to define. A SHORT INTRODUCTION TO CATEGORY THEORY 9

(4) Let I be the category with two objects A, B of C and only two morphisms f1, f2 : A −→ B of C, in addition to the identities maps. In this case the limit is called equalizer. Write down explicitly the universal property in this case. Prove that if C = Ab and f2 = 0 then the equalizer is nothing else that ker f1 equipped by the inclusion ker f1 → A. Moreover prove that any equalizer is a monomorphism. (5) Let I be the category given by a preordered set (I, ≤) (see Example 1.1.5(11)). We also suppose that it is directed, i.e. for any i, j ∈ X there exists k ∈ I such that i ≤ k and j ≤ k. In this case the limit is called and noticed by lim F (i). Write down explicitly the universal ← i∈I property in this case. Prove that if C = Sets and, for any i ≥ j, F (i) −→ F (j) is the inclusion then the direct limit is the intersection indexed by I. Exercise 1.5.6. Prove that a category C has finite products (resp. ) indexed by a non-empty set I if and only if it has the product (resp. coproducts) of two objects. Deduce that it has all finite products (resp. coproduct) if and only if it has the product (resp. coproducts) of two objects and the terminal object T (resp. initial object S). Moreover prove that these constructions are associative up to a unique isomorphism and that, for any object A, A × S ' A and A q S ' A. 1.6. Free objects. The following concept generalizes the concept of the of a vector space. Definition 1.6.1. Let (C,F ) be a concrete category. Let I be a set. A over I is an object A of C equipped by a function i : I −→ F (A) such that for any object B and any function f : I → F (B), there exists a unique morphism f˜ : A → B such that f = F (f˜) ◦ i. That is, the following diagram commutes

I i / F (A) f F (f˜) !  F (B) We remark that usually F is just the functor which associates to an object A its underlying set. The above condition simply says that to give a morphisms from a free object to another object is the same that choosing an element of F (X) for any i ∈ I. The choice of the function i corresponds essentially to the choice of the basis. Let us consider the category R-Mod, which includes of course the case of vector spaces. Exercise 1.6.2. Prove that for any set I the free object over I in the category of R-modules is RI := I n ⊕i∈I R with the obvious i : I → R . If I is finite of n we simply write R . In particular the 4 free object over I in Ab is ZI . Show that it is not a free object in Grp. . References [Ma] S.Mac Lane Categories for the working Mathematician Graduate Texts in Mathematics 5, Springer-Verlag, Second edition, 1997.