22 Categories and Functors

This sections is a review of basic concepts in category theory. It will follow as an immediate consequence of basic deﬁnitions that all left adjoint functors are right exact and similarly for right adjoint functors. Deﬁnition 22.1. A category C is a collection Ob(C) of objects A and morphism sets HomC(A, B) whose elements are written f : A → B (with A = s(f), B = t(f)) together with a composition law

c : HomC(B,C) × HomC(A, B) → HomC(A, C) written c(f, g) = f ◦ g satisfying the following.

1. Each object A ∈ C has an identity morphism idA : A → A so that idt(f) ◦ f = f = f ◦ ids(f). 2. Composition is associative when defined ((f ◦ g) ◦ h = f ◦ (g ◦ h)). Definition 22.2. A preadditive category is defined to be a category so that each morphism set HomC(A, B) is an additive group and composition is bi- additive, i.e.: (g1 + g2) ◦ f = g1 ◦ f + g2 ◦ f

g ◦ (f1 + f2) = g ◦ f1 + g ◦ f2 For any object X of any preadditive category C, the endomorphism set

EndC(X) = HomC (X,X) is a ring with multiplication given by composition of morphisms. Further- more, each morphism set HomC (X,Y ) has the structure of an End(Y )- End(X)-bimodule with the action of these two rings given by composition. Example 22.3. The category Ab = Z-Mod of all additive groups and homomorphisms is a preadditive category as is any full subcategory1 of Ab.

Example 22.4. For any ring R let CR be the preadditive category with one object ∗ and morphism set HomCR (∗, ∗) equal to the additive group R with composition given by ring multiplication. Recall that a functor F : C → D sends objects to objects and morphisms to morphisms so that F commutes with s, t, id, ◦, i.e., 1. If f : A → B then F (f): FA → FB (i.e., sF (f) = F (sf) and tF (f) = F (tf)).

2. F (idA) = idFA : FA → FA.

1A full subcategory D of a category C is given by a subcollection of Ob(C) together with the entire set of morphisms between any two elements: HomD(A, B) = HomC(A, B).

1 3. F (fg) = F (f)F (g). Deﬁnition 22.5. If C, D are preadditive categories, a functor F : C → D is called additive if F induces a homomorphism

F : HomC(A, B) → HomD(F A, F B) for all A, B ∈ Ob(C).

Lemma 22.6. An additive functor F : CR → Ab is the same as a left R- module. Proof. Suppose that M is a left R-module. Then M ∈ Ob(Ab). Let

FM : CR → Ab be given by FM (∗) = M, FM (r) = θr : M → M (the action of r on M, i.e., θr(x) = rx). Then FM is an additive functor. For example the third property is: FM (rs) = θrs = θrθs = FM (r)FM (s)

Conversely, suppose that F : CR → Ab is an additive functor. Then the associated R-module is the additive group F (∗) with the action of r ∈ R given by: rx = F (r)(x). Lemma 22.6 has several conceptual implications. First, it leads us to think about a preadditive categories C as a generalization of a ring and additive functors C → Ab as representing C-modules. The second idea is that if R-modules are represented as functors then homomorphisms of R-modules should be natural transformations of these functors.

Natural Transformations Recall that a natural transformation η : F → G between two functors F,G : C → D is a family of morphisms

(ηX : FX → GX) ∈ HomD(F X, GX) for each X ∈ Ob(C) making the following diagram commute for all morphisms f : X → Y in C. η FX −−−→X GX     F (f)y yG(f) η FY −−−→Y GY Note that if C is a small category in the sense that its collection of objects Ob(C) is a set (e.g., CR is small but Ab is not) then a natural transformation η is an element of the set Y HomD(F X, GX). X∈Ob(C) This set is an additive group if D is a preadditive category.

2 Deﬁnition 22.7 (Functor Categories). If C is a small category and D is any category let Fun(C, D) be the category whose objects are morphisms F : C → D and whose morphisms are natural transformations of such functors. If C, D are preadditive categories (with C small) let Hom(C, D) be the full subcategory of all additive functors F : C → D. Theorem 22.8. The category of all left R-modules is isomorphic to the additive functor category Hom(CR, Ab).

Adjoint Functors Definition 22.9. Adjoint functors between any two categories C, D are defined to be functors F : C → D and G : D → C so that there is a natural bijection: =∼ ηXY : HomC(GX, Y ) −→ HomD(X,FY ) for any X ∈ Ob(D) and Y ∈ Ob(C). Additive functors between preadditive categories are adjoint if ηXY is an isomorphisms of additive groups for all X and Y . Example 22.10. The forgetful functor F : Ab → Sets is right adjoint to the free abelian group functor G = Z[−]: Sets → Ab. Example 22.11. Hom(A, −) and A ⊗ − are adjoint additive functors: Hom(A ⊗ B,C) ∼= Hom(B, Hom(A, C)) The well-know fact that tensor product is right exact and Hom is left exact are special cases of the following more general theorem where the definitions are given in the proof. Theorem 22.12. All left adjoint additive functors are right exact and all right adjoint additive functors are left exact. Proof. When we say that any left adjoint functor G : D → C is right exact we mean that it takes an exact sequence:

f g A −→ B −→ C → 0 (1) to an exact sequence:

G(f) G(g) GA −−→ GB −−→ GC → 0. (2) The exactness of the sequence (1) is by deﬁnition equivalent to saying that C is the cokernel of f, i.e., g ◦ f = 0 and any morphism h : B → X in D so that h ◦ f = 0 factors uniquely through C: f g A - BC- @ @ h 0 e @@R ? © ∃!h X

3 In other words, the following sequence of abelian groups is exact for all objects X in D:

g∗ f ∗ 0 → HomD(C,X) −→ HomD(B,X) −→ HomD(A, X). (3)

To show that (2) is exact we therefore need to show that the following sequence of abelian groups is exact for all Y ∈ Ob(C):

G(g)∗ G(f)∗ 0 → HomC(GC, Y ) −−−→ HomC(GB, Y ) −−−→ HomC(GA, Y ).

However, by assumption, this sequence is equivalent to the sequence (3) in the special case when X = FY . The left exactness of the right adjoint functor F is left as an exercise.

Remark 22.13. Note that the above proof incorporates the fact that Hom is left exact. Thus it would in some sense be circular reasoning to use Theo- rem 22.12 to prove that Hom is left exact.

Theorem 22.12 is a special case of the following more general theorem (which does not require additivity). We will discuss push-outs and pull-backs later.

Theorem 22.14. Left adjoint functors preserve push-outs and right adjoint functors preserve pull-backs.