Matemaattis-Luonnontieteellinen Matematiikan Ja Tilastotieteen Laitos Joni Leino on Mitchell's Embedding Theorem of Small Abel

Home , Five lemma, Snake lemma

HELSINGIN YLIOPISTO — HELSINGFORS UNIVERSITET — UNIVERSITY OF HELSINKI Tiedekunta/Osasto — Fakultet/Sektion — Faculty Laitos — Institution — Department

Matemaattis-luonnontieteellinen Matematiikan ja tilastotieteen laitos Tekijä — Författare — Author Joni Leino Työn nimi — Arbetets titel — Title

On Mitchell’s embedding theorem of small abelian categories and some of its corollaries Oppiaine — Läroämne — Subject Matematiikka Työn laji — Arbetets art — Level Aika — Datum — Month and year Sivumäärä — Sidoantal — Number of pages Pro gradu -tutkielma Kesäkuu 2018 64 s. Tiivistelmä — Referat — Abstract

Abelian categories provide an abstract generalization of the category of modules over a unitary ring. An embedding theorem by Mitchell shows that one can, whenever an abelian category is suﬃciently small, ﬁnd a unitary ring such that the given category may be embedded in the category of left modules over this ring. An interesting consequence of this theorem is that one can use it to generalize all diagrammatic lemmas (where the conditions and claims can be formulated by exactness and commutativity) true for all module categories to all abelian categories.

The goal of this paper is to prove the embedding theorem, and then derive some of its corollaries. We start from the very basics by defining categories and their properties, and then we start con- structing the theory of abelian categories. After that, we prove several results concerning functors, "homomorphisms" of categories, such as the Yoneda lemma. Finally, we introduce the concept of a Grothendieck category, the properties of which will be used to prove the main theorem. The final chapter contains the tools in generalizing diagrammatic results, a weaker but more general version of the embedding theorem, and a way to assign topological spaces to abelian categories. The reader is assumed to know nothing more than what abelian groups and unitary rings are, except for the final theorem in the proof of which basic homotopy theory is applied.

Avainsanat — Nyckelord — Keywords Category theory, abelian categories, homological algebra Säilytyspaikka — Förvaringsställe — Where deposited Kumpulan tiedekirjasto

Muita tietoja — Övriga uppgifter — Additional information On Mitchell’s embedding theorem of small abelian categories and some of its corollaries

Joni Leino Contents

Introduction 2

1 Category theoretical preliminaries 4

2 Abelian categories 10

3 Module categories 26

4 On functors 30

5 On the Grothendieck property and injectivity 42

6 Two embedding theorems 49

7 Some corollaries of the embedding theorems 58

8 References and bibliography 63

1 Introduction

On one hand, studying homological algebra involves a lot of category theoretical constructions. On the other hand, some of its results, namely the ones involving exactness and commutativity in a diagram, can be generalized to categories under certain conditions. As it is not always the case that the objects in these categories have underlying sets, the methods familiar from homological algebra involving the element relation of set theory will not be of use in these generalizations, and this will make proving the results harder inside these categories. Luckily, a way to bypass such difficulties exists. It is easily motivated by a topological analogy. Manifolds, an important category of topological spaces, are approachable using open coverings called atlases. These atlases consist of so called chart neighbourhoods, or open sets homeomorphic to a euclidean space of some dimension. These atlases lead to two pleasant things. The first of these is that, due to the homeomorphism, whatever local properties (for example local compactness or local connectedness) hold in euclidean spaces, they also hold for all manifolds, and the second one is that the mathematician studying a manifold can transfer points from the possibly quite abstract manifold to a space much more familiar and easy to deal with by using these local homeomorphisms. Of course, this comes with the limitation that it need not be possible to consider the whole manifold at once like this, but instead it may be restricted to strict subsets. An interesting thing is that, although seemingly different, this same idea can be used while dealing with abelian categories, or a special type of categories sharing a lot of properties with the category of abelian groups. This is based on the main result of this text, the embedding theorem by Mitchell, and several smaller results. On the way to Mitchell’s theorem, we will prove that any set of objects in an abelian category gives rise to a small abelian category, and this will serve as an analogue for the chart neighborhoods of points. We will then use the claim of the embedding theorem: for every small abelian category, one can always find a unitary ring such that the category can be embedded into the category of left modules over this ring, and these categories will be the "euclidean spaces" in this context. Applying the metatheorems, according to which diagrammatic lemmas hold in abelian categories if and only if they hold in the categories of abelian groups or these module categories, one obtains an analogue to the local inheritance of compactness and connectedness. Small categories will mean that the class of objects in the given category is a set instead of a proper class. Without spending time in set theoretic considerations, we will use the von Neumann-Bernays-Gödel axioms, so we

2 can speak of the category of sets, groups, topological spaces, and so on, but not of the category of all categories. Due to the simplicity of the examples in this text, it will be sufficient to divide classes into two cases, namely the ones with and the ones without a cardinality, and these will be enough to function as sets and proper classes, respectively. The requirement of smallness will act as a limiting factor, just like some manifolds need to be considered piecewise instead of being homeomorphic to euclidean spaces as whole. Mitchell’s theorem does not allow embedding an arbitrary abelian category at once, but "locally" using subcategories de- termined by choosing a set of objects. It is also possible to generalize the result, but this will also weaken it. In the final section, we will consider abelian categories equivalent to small abelian categories, and induce equiv- alences between the so called essentially small category and a subcategory of a module category. A related example will show that this indeed does generalize Mitchell’s theorem, but does not cover all abelian categories. The text, which is heavily based on Freyd’s book on abelian categories, will be divided into seven sections. The first two will construct the basic theory of general categories and then abelian categories. The third section will be used to prove that the category of left modules over a unitary ring is indeed an abelian category. In the fourth section, we will derive basic facts regarding functors, as these will then serve as tools needed later, and we will prove a sufficient condition for the desired kind of embeddings to exist. In the fifth section, we will develop more necessary tools to be used in the sixth section where the embedding theorem is finally proved. The seventh section will be on several consequences of the theorem by Mitchell, such as the generalization of diagrammatic lemmas, or the existence of a classifying space unique up to weak homotopy equivalence for suitable abelian categories. Finally, I would like to thank Marja Kankaanrinta and Pekka Pankka for their valuable feedback and help.

3 1 Category theoretical preliminaries

The goal of this section is to lay down the basic category theoretical con- cepts we will be using later. We begin by deﬁning categories, subcategories and opposite categories, and then we deﬁne several commonly encountered properties.

Definition 1.1. A category C consists of (1) A class of objects obj C; (2) For any objects A, B ∈ obj C, a class HomC(A, B) of morphisms A → B; (3) A binary operation ◦ for morphisms such that: (i) g◦f ∈ HomC(A, C) is defined iff f ∈ HomC(A, B) and g ∈ HomC(B,C) for some A, B, C ∈ obj C; (ii) Whenever h◦(g◦f) is defined, so is (h◦g)◦f, and h◦(g◦f) = (h◦g)◦f; (iii) For every A, B ∈ obj C, there are morphisms 1A, 1B called the identities of A and B, respectively, such that 1B ◦ f = f = f ◦ 1A for every f ∈ HomC(A, B). Lemma 1.2. For any category C, the identities are unique.

0 Proof. Let A be an object in a category C. If now 1A, 1A are identities of A, then 0 0 1A = 1A ◦ 1A = 1A.

Deﬁnition 1.3. A subcategory C’ of a category C consists of (1) a subclass obj C’ of obj C; (2) for all A, B ∈ obj C’, a subclass HomC’(A, B) of HomC(A, B); (3) the composition ◦ of C such that: (i) if A ∈ obj C’, then 1A ∈ HomC’(A, A); (ii) if f ∈ HomC’(A, B) and g ∈ HomC’(B,C), then g◦f ∈ HomC’(A, C). If HomC’(A, B) = HomC(A, B) for all A, B ∈ obj C’, then C’ is a full subcategory of C.

We will give a short list of examples of categories now. The ﬁfth example will be of great importance later when we discuss the abelian categories, an example of which it is. (i) The category Sets with sets as its objects, functions as its morphisms and the ordinary composition of functions as its composition.

4 (ii) The category Top with topological spaces as its objects, continuous maps as its morphisms and the ordinary composition. (iii) The category Haus with Hausdorﬀ spaces as its objects is a subcategory of Top. (iv) The category Grp of groups with groups as its objects, group homomorphisms as its morphisms and the ordinary composition of functions. (v) The subcategory Ab of Grp with abelian groups as its objects.

Deﬁnition 1.4. The opposite category Cop of a category C is formed by reversing each morphism in C and the order of composition. That is, if op op op op f ∈ HomC(A, B), then f ∈ HomCop (B,A), and (g ◦ f) = f ◦ g , when deﬁned.

For example, consider the categories Sets and Setsop, and sets ∅ and X 6= ∅. In Sets, there is an inclusion function i: ∅ → X, so there is the dual morphism iop : X → ∅ in Setsop, but now iop can not be a function.

Deﬁnition 1.5. Let C be a category, and let σ be a statement regarding C. Form the dual statement σop of σ by replacing all the morphisms involved in σ with their corresponding morphisms in Cop. If σ = σop, then σ is self-dual.

Deﬁnition 1.6. Let C be a category, let A, B, E, M ∈ obj C, and let f : A → B be any morphism. Now, f is a monomorphism (in C) if, for any object M and any morphisms i, j : M → A, it is true that f ◦i = f ◦j implies i = j, and f is an epimorphism (in C) if, for any object E and any morphisms p, q : B → E, it holds that p ◦ f = q ◦ f implies p = q.

Deﬁnition 1.7. Objects A and B are isomorphic if there are morphisms f : A → B and g : B → A such that g ◦ f = 1A and f ◦ g = 1B. We denote this by A ∼= B. Lemma 1.8. (1) In a category, the composites of monomorphisms are monomorphisms, the composites of epimorphisms are epimorphisms, and the composites of isomorphisms are isomorphisms. (2) If f, g are such morphisms in a category that their composite g ◦ f is deﬁned, and this composite is an isomorphism, then f is a monomorphism and g is an epimorphism. In particular, all isomorphisms are monomorphisms and epimorphisms. (3) The notions of monomorphism and epimorphism are dual, and the notion of isomorphism is self-dual.

Proof. (1) Immediate from the deﬁnitions.

5 (2) Suppose f : A → B and g : B → C are such that g ◦ f is an isomorphism. Let i, j : M → A be such that f ◦ i = f ◦ j. Now,

(g ◦ f) ◦ i = g ◦ (f ◦ ((i) = g ◦ (f ◦ j) = (g ◦ f) ◦ j, by associativity, so now i = j because g◦f was assumed to be an isomorphism. Similarly, if p, q : C → E are such that p ◦ g = q ◦ g, then

p ◦ (g ◦ f) = (p ◦ g) ◦ f = (q ◦ g) ◦ f = q ◦ (g ◦ f), and thus p = q. By deﬁnition, f is now a monomorphism and g an epimorphism. (3) Immediate.

Deﬁnition 1.9. Let C be a category. An object D in C is an initial object (of C) if, for any C ∈ obj C, there is a unique morphism D → C.

Lemma 1.10. The initial objects of a category C are unique up to isomorphism.

Proof. Suppose D is an initial object of C. If C is another initial object, then there are unique morphisms C → D and D → C, so there are unique morphisms C → D → C and D → C → D. Now, by uniqueness, it must be that C → D → C = 1C and D → C → D = 1D, so it follows that C is isomorphic to the initial object D.

Deﬁnition 1.11. A terminal object in a category C is an object C such that there is a unique morphism D → C for every D ∈ obj C.

Lemma 1.12. Terminal objects in a category C are unique up to isomorphism.

Proof. Follows from the uniqueness of initial objects by duality.

Deﬁnition 1.13. Let C be a category, and let {Aj | j ∈ J} be a family of objects of C.A product of the objects Aj is an object P in C with morphisms pj : P → Aj such that, for any object C and any morphisms fj : C → Aj, there is a unique h: C → P such that the diagram

6 C h P

fj pj Aj commutes for each j ∈ J. In particular, a product of A, B ∈ obj C is an object P ∈ obj C together with morphisms pA : P → A, pB : P → B such that, for any morphisms f : C → A, g : C → B in C, there is a unique h: C → P such that the diagram

C f g h A P B pA pB commutes.

Lemma 1.14. Products are unique up to isomorphism.

Proof. Let {Aj | j ∈ J} be a family of objects in a category C with small products (that is, products exist for all sets of objects), and let P , Q be products of the family, and let pj, qj be the morphisms in the deﬁnition of a product. By deﬁnition, there are morphisms f : P → Q, g : Q → P such that the diagrams P Q pj qj f g Q A P A qj j pj j g f pj qj P Q commute. By uniqueness, it must be that g ◦ f = 1P and f ◦ g = 1Q.

Deﬁnition 1.15. Let C be a category, and let {Aj | j ∈ J} be a family of objects of C.A coproduct of the objects Aj is an object C in C with morphisms λj : Aj → C such that, for any object X and any morphisms fj : Aj → X, there is a unique h: C → X such that the diagram

C h X

λj fj Aj

7 commutes for each j ∈ J. In particular, a coproduct of A, B ∈ obj C is an object C ∈ obj C together with morphisms λA : A → C, λB : B → C such that, for any morphisms f : A → X, g : B → X in C, there is a unique h: C → X such that the diagram

A λA C λB B

h f g X commutes.

Corollary 1.15.1. Coproducts are unique up to isomorphism.

We denote products by A × B and coproducts, or sums, by A ⊕ B.

Deﬁnition 1.16. Monomorphisms f : A1 → B and g : A2 → B are equivalent in a category C if there are morphisms h1 : A1 → A2 and h2 : A2 → A1 in C such that f = g ◦ h1 and g = f ◦ h2. Deﬁnition 1.17. A subobject of B in a category C is an equivalence class of monomorphisms into B in C. A subobject represented by f : A1 → B is contained in that represented by g : A2 → B if there is a morphism h: A1 → A2 such that g ◦ h = f.

Let A1 ⊂ A2 ⊂ B be sets, and let i: A1 → B, j : A2 → B be inclusions. If k : A1 → A2 is an inclusion, then i = j ◦ k, so the deﬁnition makes sense. Moreover, containment induces a partial ordering in the class of subobjects of any ﬁxed object in a category.

Deﬁnition 1.18. Epimorphisms f : A → B1 and g : A → B2 are equivalent in a category C if there are morphisms h1 : B1 → B2 and h2 : B2 → B1 in C such that f = h2 ◦ g and g = h1 ◦ f. Deﬁnition 1.19. A quotient object is an equivalence class of epimorphisms. The quotient object represented by f : A → B1 is smaller than the quotient object represented by g : A → B2 if there is a morphism h: B2 → B1 such that f = h ◦ g.

From now on, when we speak of subobjects of a given object, by "the subobject such that... ", we mean the equivalence class of monomorphisms with the desired property, and by "a subobject such that... " we mean some object such that there exists a monomorphism from this object to the given one or a monomorphism between them. We will treat quotient objects similarly.

8 Deﬁnition 1.20. Let f : A → B be a morphism in a category. An image of f is a monomorphism i: B0 → B such that (1) There exists a morphism f 0 : A → B0 such that f = i ◦ f 0. (2) For any object C with a morphism f : A → C and a monomorphism j : C → B such that f = j ◦ f, there is a unique m: B0 → C such that the diagram

f A B f 0 i

B0 f j m C commutes.

Lemma 1.21. Images (if they exist) are unique up to isomorphism.

We will prove later that all morphisms have images in abelian categories. Proof. Let f : A → B be an arbitrary morphism, and let i: B0 → B, j : B → B be images of f. Consider the diagram

f A B f 0 i

0 f B j

f 0 i B

B0 in which i, j are monomorphisms, i ◦ f 0 = f = j ◦ f, and m, n are the unique morphisms such that i = j ◦ m, j = i ◦ n. Now, i = j ◦ m = (i ◦ n) ◦ m = i◦(n◦m), so, by uniqueness, n◦m = 1B0 , and, similarly, m◦n = 1B, proving the claim. We introduce the notation im f for an image of f.

9 2 Abelian categories

In this section we will deﬁne abelian categories, and prove certain results for them to be used later. Deﬁnition 2.1. A zero object 0 in a category A is an object that is both initial and terminal in A.A zero morphism f : A → B in A is a morphism such that the diagram

f A B

0 commutes. Definition 2.2. A category A is preadditive if, for any objects A, B in A, Hom(A, B) is an abelian group (with an operation denoted by +) and the composition of morphisms is bilinear, that is, (f + g) ◦ h = f ◦ h + g ◦ h and h ◦ (f + g) = h ◦ f + h ◦ g when defined. A category A is additive if it is preadditive and has all finite sums. Definition 2.3. Given any f, g : A → B in a category A, (K, k), where K ∈ obj A and k : K → A, is a difference kernel of f and g if (1) f ◦ k = g ◦ k, (2) for all h: X → A such that f ◦ h = g ◦ h, there is a unique u: X → K such that the diagram

X u h

K A k commutes. This is a kernel of f if g is a zero morphism. Deﬁnition 2.4. Given any f, g : A → B in a category A, (C, c), where C ∈ obj A and c: B → C, is a diﬀerence cokernel of f and g if (1) c ◦ f = c ◦ g, (2) for all h: B → X such that h ◦ f = h ◦ g, there is a unique u: C → X such that the diagram

B c C

h u X

10 commutes. This is a cokernel of f if g is a zero morphism.

Lemma 2.5. Kernels and cokernels are unique up to isomorphism if they exist.

Proof. Suppose f : A → B is a morphism in a category such that it has kernels k : K → A, k0 : K0 → A. Now we obtain a diagram K

0 ϕ k

0 k0 f 1K K A B

θ k 0

K where ϕ: K → K0 is the unique morphism such that k0 ◦ϕ = k, and similarly θ : K0 → K is the unique morphism such that k ◦ θ = k0. Now, k ◦ (θ ◦ ϕ) = (k ◦ θ) ◦ ϕ = k0 ◦ ϕ = k, so ψ = θ ◦ ϕ is the unique morphism such that k ◦ ψ = k, so ψ = 1K . One obtains ϕ ◦ θ = 1K0 in a similar way. The claim for cokernels follows by dualizing the arguments.

Lemma 2.6. Whenever they exist, kernel morphisms are monomorphisms and cokernel morphisms are epimorphisms.

Proof. We will prove the claim for kernels, and the proof for cokernels will follow from duality. Let k : K → A be a kernel of some morphism f : A → B in a category with kernels, and let i, j : M → K be such that k ◦i = k ◦j = g. Now f ◦ g = 0, so there is a unique ϕ: M → K such that k ◦ ϕ = g, but this implies ϕ = i = j, so k is a monomorphism.

Deﬁnition 2.7. An additive category A is abelian if (A0) it has a zero object, (A1) for every pair of objects, there is a product, (A1*) for every pair of objects, there is a sum, (A2) for every morphism, there is a kernel, (A2*) for every morphism, there is a cokernel, (A3) every monomorphism is a kernel of a morphism, (A3*) every epimorphism is a cokernel of a morphism.

11 Since all the axioms Ai, i = 1, 2, 3, are dual to the axioms Ai∗ and A0 is self-dual, we see that A is abelian if and only if Aop is. As we have already stated, the category of abelian groups is abelian. Another example is obtained if one considers the category of chain complexes of abelian groups. A chain complex A∗ = (An, dn)n∈Z consists of abelian groups An, n ∈ Z, and homomorphisms dn : An → An−1 such that dn ◦dn+1 = 0 for every n ∈ Z, and the morphisms f∗ : A∗ → B∗ = (Bn, δn)n∈Z are families {fn : An → Bn | n ∈ Z} of homomorphisms such that the diagram

dn An An−1

fn fn−1

Bn Bn−1 δn commutes for every n ∈ Z. The sums are given by A∗ + B∗ = (An ⊕ Bn, dn + δn)n∈Z, where An ⊕ Bn is the direct sum, and so this gives also the finite products A∗ × B∗ = A∗ + B∗. Choosing An = 0 for every n ∈ Z gives a zero object, and the rest of the axioms follow from the corresponding properties of abelian groups. We conclude that the category CompAb of chain complexes of abelian groups is an abelian category, assuming Ab is. This will be proved in the next section. Consider an object A in an abelian category A, and let SA,QA be the classes of subobjects and quotient objects of A, respectively. Define functions K : QA → SA and C : SA → QA to be such that K assigns kernels and C assigns cokernels. Using these we will get a important result, Theorem 2.9. Lemma 2.8. Using the definitions above, K and C are inverse functions. Proof. Let i: A0 → A be a monomorphism. By axiom (A3), it is a kernel of some morphism f : A → B. Let p: A → F be a cokernel of i, and let k : A00 → A be a kernel of p. Since f ◦ i = 0, there is a unique morphism g : F → B such that the diagram A0 F i p

A g

k f A00 B commutes. Now, we note that f ◦ k = g ◦ p ◦ k = 0, so there is a morphism ϕ: A00 → A0 such that i ◦ ϕ = k. Also, p ◦ i = 0, so there is some morphism ψ : A0 → A00 such that k ◦ ψ = i. Now the monomorphisms i and k are

12 equivalent, so they represent the same subobject of A. It follows that K ◦ C is the identity. Dualizing the arguments, one obtains the same for C ◦K.

Theorem 2.9. A morphism in an abelian category is an isomorphism if and only if it is both an epimorphism and a monomorphism.

Proof. Let f : A → B be both a monomorphism and an epimorphism. Then B → 0 is a cokernel of f, and 1B is clearly a kernel of B → 0, it follows from the previous lemma that so is f, and hence A ∼= B, as kernels are unique up to isomorphism. The other direction follows from Lemma 1.8.

Deﬁnition 2.10. The intersection of two subobjects of an object A is their greatest lower bound in the class of subobjects of A.

Lemma 2.11. In an abelian category, every pair of subobjects of an object A has an intersection.

Proof. Let i: A1 → A, j : A2 → A be monomorphisms, p: A → F a cokernel of i and k : A12 → A2 a kernel of p ◦ j. Since the composite p ◦ j ◦ k is zero, 0 we obtain a unique k : A12 → A1 such that the diagram k A12 A2

k0 j A A 1 i commutes, since i is a kernel of p by Lemma 2.6. Let then f : X → A1, g : X → A2 be any morphisms such that i ◦ f = j ◦ g. We claim there exists a unique 0 h: X → A12 such that k ◦ h = f, k ◦ h = g. The existence follows from the fact that p ◦ j ◦ g = p ◦ i ◦ f = 0, so we can use the fact that k is a kernel of p ◦ j, and now i ◦ k0 ◦ h = j ◦ k ◦ h = j ◦ g = i ◦ f. 0 0 If now α, β : M → A12 are any morphisms such that k ◦α = k ◦β, then we obtain j ◦k◦α = i◦k0 ◦α = i◦k0 ◦β = j ◦k◦β. Since j, k are monomorphisms, so is their composite, and we obtain α = β. We have now obtained that A12 is a subobject of both A1 and A2. Now, suppose f : X → A1 represents a subobject of both A1 and A2. Then there exists some g : X → A2 such that i ◦ f = j ◦ g, and so we obtain a morphism h: X → A12, as proved above. This morphism will be a monomorphism, since if α, β : M → X satisfy h◦α = h◦β, then f ◦α = k0 ◦h◦α = k0 ◦h◦β = f ◦β, so α = β. This proves that h: X → A12 represents a subobject of A12, and so A12 represents a the greatest lower bound of A1,A2 in the class of subobjects of A. Dualizing the arguments, one obtains that every pair of subobjects of A has a union, that is, their least upper bound, in the class of subobjects of A.

13 We introduce the notation A1 ∪A2 for a union and A1 ∩A2 for an intersection of A1 and A2. Lemma 2.12. In an abelian category, every morphism f : A → B has an image. Moreover, as a subobject of B, the image satisfies im f = (K ◦C)(f), where K,C are the kernel and cokernel assigning functions defined above. Proof. We will say that a monomorphism s: S → B allows f if there is a morphism t: A → S such that s ◦ t = f, and that an epimorphism p: B → F kills f if p ◦ f = 0. We shall show that a subobject s: S → B allows f if and only if its cokernel kills f. It will then follow that C(f) will be the largest quotient object that kills f, and so (K ◦ C)(f) will be the smallest subobject allowing f. When this is done, the claim will follow from the definition of an image. Suppose s allows f, and let p: B → F be a cokernel of s. Then p ◦ f = p ◦ s ◦ t = 0, for some t: A → S, so the cokernels of s kill f. Conversely, suppose that the cokernels of a monomorphism s: S → B kill f, and let p: B → F be one of these. Since s is now a kernel of p, there is a morphism t: A → B such that f = s ◦ t, so s allows f. Definition 2.13. For a morphism f : A → B in an abelian category, the coimage of f is the smallest quotient object of A through which f factors. Lemma 2.14. For any morphism f : A → B in any abelian category, we have coim f = (C ◦ K)(f). Proof. Since coimages are dual to images, the claim follows from Lemma 2.12 by duality. Lemma 2.15. In an abelian category, a morphism f : A → B is an epimorphism if and only if, as subobjects of B, the equality im f = B holds, and hence if and only if coker f = 0. Proof. Suppose coker f = 0, so im f = B, the subobject represented by 1B. Suppose x, y : B → C are morphisms such that x ◦ f = y ◦ f, and let k : ker (x−y) → B be the difference kernel of x and y, where ker (x−y) is the kernel of x − y ∈ Hom(B,C). Then there is a morphism g : A → ker (x − y) such that k ◦ g = f, and so ker (x − y) contains im f. Thus, we obtain ker (x−y) = B, and so x = y. Therefore f is an epimorphism. The converse direction is clear. Corollary 2.15.1. In an abelian category: (1) For any morphism f : A → B, there exists an epimorphism p: A → im f. (2) A morphism f : A → B is a monomorphism if and only if ker f = 0, or equivalently coim f = A.

14 Proof. For the ﬁrst claim, suppose C(p) 6= 0, for every p: A → im f. Then p factors through a proper subobject of C(p), contradicting the minimality of images. The second claim follows from 2.15 by duality. Deﬁnition 2.16. Let A be an abelian category. Let A, B, C ∈ obj A and f ∈ Hom(A, B), g ∈ Hom(B,C). The sequence

f g A B C is exact at B if im f = ker g. The sequence

f g 0 A B C 0 is a short exact sequence if it is exact at A, B, C. If Aj ∈ obj A and fj : Aj → Aj−1 for all j ∈ Z, then the sequence

fj+1 fj ··· Aj+1 Aj Aj−1 ··· is exact if it is exact at each Aj. Lemma 2.17. For any morphism f : A → B, we have an exact sequence

f p 0 K k A B C 0 in which k : K → A is a kernel and p: B → C is a cokernel of f. Moreover, f is a monomorphism if and only if K = 0, an epimorphism if and only if C = 0 and an isomorphism if and only if K = 0 = C. Proof. Since im k = (K ◦ C)(k) = K(f) = ker f, we have the exactness at A. Similarly, ker p = K(p) = (K ◦ C)(f) = im f, so we have exactness at B. We will show that the sequence is exact at K, as the arguments for C are dual. Since k : K → A is a monomorphism, its kernels must be zero. We claim that the image of o: 0 → K is zero as well, but this follows by noting that there is an epimorphism 0 → im o, and that the unique morphism o can be an epimorphism only when there exists at most one morphism from im o to any object in the category. On the other hand, we can always obtain a morphism between any two objects in an abelian category by taking the zero morphism, so the image of o must be initial. We claim it is also terminal. Let x, y : X → im o be any morphisms. If i: im o → K is the monomorphism in the deﬁnition of an image, then x 6= y implies 0 = o◦(X → 0) = i◦x 6= i◦y = o◦(X → 0) = 0, and so we conclude that the image of o is a zero object, and this gives exactness at K. The claim for monomorphisms follows from Corollary 2.15.1 (2), the claim for epimorphisms from Theorem 2.15, and the claim for isomorphisms from Theorem 2.9.

15 Deﬁnition 2.18. A commutative diagram

g P B

f ψ

A ϕ C is a pullback diagram in a category C if, for every object X and every pair of morphisms α: X → A, β : X → B in C such that ϕ ◦ α = ψ ◦ β, there is a unique h: X → P (in C) such that the diagram

X h β

g P B α f ψ

A ϕ C commutes.

For example, consider the category of sets and functions. If f : A → C and g : B → C are any functions, one can deﬁne a set

Pfg := {(a, b) ∈ A × B | f(a) = g(b)}.

If p: Pfg → A and q : Pfg → B are deﬁned by (a, b) 7→ a, (a, b) 7→ b, respectively, one obtains a commutative diagram

q Pfg B

p g A C. f

If then X is any set and ϕ: X → A, ψ : X → B are functions such that f ◦ ϕ = g ◦ ψ, one can deﬁne θ : X → Pfg by x 7→ (ϕ(x), ψ(x)). Clearly, p ◦ θ = ϕ, q ◦ θ = ψ, so we have established the existence of an arrow like the 0 dashed one in the deﬁnition of pullbacks. If now θ : X → Pfg was any arrow giving commutativity, we would have ((p, q)◦θ0)(x) = ((p◦θ0)(x), (q◦θ0)(x)) = (ϕ(x), ψ(x)) = ((p ◦ θ)(x), (q ◦ θ)(x)) = ((p, q) ◦ θ)(x), for every x ∈ X, but 0 (p, q) = 1Pfg , so θ = θ, proving the uniqueness. It follows that the diagram above is a pullback diagram.

16 Theorem 2.19. Every diagram

A ϕ C in an abelian category A can be enlarged to a pullback diagram. Moreover, pullbacks are unique up to isomorphism.

Proof. Consider P = A × B and arrows p1 : P → A, p2 : P → B. Let (K, k) be a diﬀerence kernel of ϕ ◦ p1, ψ ◦ p2, and deﬁne k1 = p1 ◦ k, k2 = p2 ◦ k. Let X be any object, and let α, β be any morphisms such that the diagram X β

K k2 B α

k1 ψ

A ϕ C commutes. Now, there is a unique morphism h: X → P such that p1 ◦ h = 0 α, p2 ◦h = β, so now (ϕ◦p1)◦h = (ψ◦p2)◦h, and there is a unique h : X → K 0 0 0 such that k ◦ h = h. Then α = p1 ◦ h = p1 ◦ k ◦ h = k1 ◦ h , β = p2 ◦ h = 0 0 p2 ◦ k ◦ h = k2 ◦ h . Thus, the diagram K k2 B

k1 ψ

A ϕ C is a pullback diagram. Let then Q, Q0 be such that the diagrams g Q B

f ψ

A ϕ C and

17 g0 Q0 B

f 0 ψ

A ϕ C are pullback diagrams. Now, consider the diagram Q

g Q0

g0 f f 0 Q g B

f ψ

A ϕ C. One can easily see it is commutative when h: Q0 → Q is such that f 0 = f ◦ h and g0 = g◦h, and, similarly, h0 : Q → Q0 is such that f 0◦h0 = f and g0◦h0 = g. 0 0 0 Now f = f ◦ (h ◦ h ), g = g ◦ (h ◦ h ), so, by uniqueness, h ◦ h = 1Q. Similarly, 0 ∼ 0 one obtains h ◦ h = 1Q0 , so Q = Q . Deﬁnition 2.20. A commutative diagram

ψ C B ϕ g A P f is a pushout diagram in a category C if, for every object X and every pair of morphisms α: A → X, β : B → X in C such that α ◦ ϕ = β ◦ ψ, there is a unique h: P → X (in C) such that the diagram

ψ C B

ϕ g

β A P f

h α

18 commutes.

Theorem 2.21. Every diagram

ψ C B ϕ A in an abelian category A can be enlarged to a pushout diagram. Moreover, pushouts are unique up to isomorphism.

Proof. Dual to the previous theorem.

Theorem 2.22. Let

f P A

g g0 B C f 0 be a pullback diagram, and k : K → P a kernel of g in an abelian category. Then f ◦ k is a kernel of g0. In particular, g is monomorphic iﬀ g0 is monomorphic.

Proof. Suppose h: X → A is such that g0 ◦ h = 0. Then the diagram X h A

0 g0 B C f 0 commutes, and so there is a unique morphism h0 : X → P such that the diagram X

h0 h

f 0 P A

g g0

B C f 0

19 commutes. This gives a unique morphism k0 : X → K such that f ◦k◦k0 = h, proving the claim.

Lemma 2.23. Given a diagram

f C A

g g0 B P f 0 in an abelian category, consider the sequence

f+g g0−f 0 C A ⊕ B P, where (g0 − f 0) ◦ (f + g) = g0 ◦ f − f 0 ◦ g. Then the following claims hold: (1) (g0 − f 0) ◦ (f + g) = 0 if and only if the diagram commutes. (2) The sequence

f+g g0−f 0 0 C A ⊕ B P is exact if and only if the square is a pullback. (3) The sequence

f+g g0−f 0 C A ⊕ B P 0 is exact if and only if the square is a pushout. (4) The sequence

f+g g0−f 0 0 C A ⊕ B P 0 is exact if and only if the square is both a pullback and a pushout.

Proof. The ﬁrst claim is obvious, the fourth claim follows immediately from the second and third claim whose proofs are similar. Therefore, only the second claim will be proven. Suppose that the square is a pullback, and the commutativity gives us that the composites are zero morphisms. Next, let h: D → A ⊕ B be any morphism in A such that (g0 −f 0)◦h = 0. The morphism h is now of the form 0 0 h = h1 + h2 with h1 : D → A, h2 : D → B, so we get (g − f ) ◦ (h1 + h2) = 0. Since the the square is a pullback, we now get a unique k : D → C such that the diagram

20 D

k h1

f C A h2

g g0

B P f 0 commutes, but now h1 = f ◦ k, h2 = g ◦ k, so f + g is a kernel morphism of g0 − f 0, and therefore also a monomorphism. To see that ker (g0−f 0) is a subobject of im (f +g), let k : K → A⊕B be a 0 0 kernel morphism of g −f . Then k = k1+k2 for some k1 : K → A, k2 : K → B, 0 0 0 0 0 0 and we have (g − f ) ◦ k = (g − f ) ◦ (k1 + k2) = g ◦ k1 − f ◦ k2 = 0, so 0 0 g ◦ k1 = f ◦ k2 and the diagram K

h k1

f C A k2

g g0

B P f 0 of solid arrows commutes. The pullback property gives the existence of a morphism h: K → C such that k1 = f ◦ h, k2 = g ◦ h, so k = (f + g) ◦ h. Now, let p: D → im (f + g) and i: im (f + g) → A ⊕ B be an epimorphism and a monomorphism, respectively, such that the diagram K k A ⊕ B i h

C p im (f + g) commutes. Now, the subobject represented by k is contained in that represented by i, but these are ker (g0 − f 0) and im (f + g), respectively. Conversely, suppose that the sequence is exact. Then (g0−f 0)◦(f +g) = 0, so the square commutes. We can then let h1 : D → A, h2 : D → B be any 0 0 0 0 morphisms in A such that g ◦ h1 = f ◦ h2, but then (g − f ) ◦ (h1 + h2) = 0.

21 Thus, there is a unique k : D → C such that (f + g) ◦ k = h1 + h2, but now f ◦ k = h1, g ◦ k = h2, so the square is a pullback. Corollary 2.23.1 (The pushout theorem). Let

C a A

b b0 B P a0 be a pushout diagram in an abelian category. If a is a monomorphism, then so is a0. Proof. The sequence

0 0 C a+b A ⊕ B b −a P 0 is exact. Now, let f, g : D → C be morphisms in the category, and suppose (a + b) ◦ f = (a + b) ◦ g. If p1 : A ⊕ B → A is the associated morphism of the product, then a = p1 ◦ (a + b), and so we have a ◦ f = (p1 ◦ (a + b)) ◦ f = p1 ◦ ((a + b) ◦ f) = p1 ◦ ((a + b) ◦ g) = (p1 ◦ (a + b)) ◦ g = a ◦ g, but a is a monomorphism, so f = g, and so a + b is a monomorphisn, and we have an exact sequence

0 0 0 C a+b A ⊕ B b −a P 0. It follows that the square is also a pullback diagram. The claim now follows from Lemma 2.11. Corollary 2.23.2 (The pullback theorem). Let

C a A

b b0 B P a0 be a pullback diagram in an abelian category. If b0 is an epimorphism, then so is b. Proof. Dual to the previous result. Deﬁnition 2.24. A category is well-powered if, for every object, the class of subobjects is a set. Deﬁnition 2.25. Let C be a category with pullbacks, and let f : A → B be any morphism and let j : B0 → B be any subobject of B in C.A preimage of B0 under f is an object C such that there is a pullback diagram

22 f 0 C B0

i j A B, f where i is a monomorphism.

It follows that C is unique up to isomorphism, so we introduce the notation f −1B0 for a preimage of B0 under f.

Lemma 2.26. Suppose that the commutative diagram

α B11 B12 ϕ β δ 0 B B B 21 γ 22 ψ 23 is such that the bottom row is exact. Then the square

α B11 B12

β δ

B21 γ B22 is a pullback if and only if the sequence

α ϕ 0 B11 B12 B23 is exact.

Proof. Suppose the square is a pullback, and let f : X → B12 be any morphism such that ϕ ◦ f = 0. Since (ψ ◦ δ) ◦ f = ψ ◦ (δ ◦ f) = 0 and γ is a kernel of ψ, there is a unique g : X → B21 such that γ ◦ g = δ ◦ f, so we can use the pullback property to obtain a unique h: X → B11 such that α ◦ h = f, so α is a kernel of ϕ. Conversely, suppose

α ϕ 0 B11 B12 B23 is exact, and let

f X B12

g δ

B21 γ B22

23 be commutative. Since ϕ ◦ f = ψ ◦ δ ◦ f = ψ ◦ γ ◦ g = 0, there is a unique h: X → B11 such that f = α ◦ h. The square is a pullback if we can show g = β ◦h, but this follows from the fact that γ ◦β ◦h = δ◦α◦h = δ◦f = γ ◦g, and γ is a monomorphism, so g = β ◦ h as desired. Lemma 2.27. If f : A → B, g : B → C are morphisms in an abelian category and g is a monomorphism, then f and g ◦ f have the same kernels. Proof. For any α: X → A, f ◦ α = 0 if and only if g ◦ f ◦ α = 0. Lemma 2.28. Consider the commutative diagram 0

α β 0 B0 B1 B2 0

1B0 1B1 γ 0 B B B 0 α 1 γ◦β 3 in which the top row is exact. Then the bottom row is exact if and only if the column is exact. Proof. Suppose the column is exact. Then, by the previous lemma, α is a kernel of γ ◦ β if it is a kernel of β, but this follows from the exactness of the top row. Thus, the bottom row is exact. Conversely, suppose the bottom row is exact, and consider the commutative diagram 0

ϕ P K 0

ψ k α β 0 B0 B1 B2 0

1B0 1B1 γ 0 B B B 0 α 1 γ◦β 3 in which the two bottom rows are exact and k : K → B2 is a kernel of γ. Then the rightmost column is exact, and the diagram ϕ P K

ψ k B B 1 β 2

24 is a pullback. By Corollary 2.23.2, the top row is exact. The goal is to show that K = 0, and we will do this by showing that k ◦ ϕ = 0. Since γ ◦ β ◦ ψ = γ ◦ k ◦ ϕ = 0, since k is a kernel of γ, there is a θ : P → B0 such that ψ = α ◦ θ, and so k ◦ ϕ = β ◦ ψ = β ◦ α ◦ θ = 0 by exactness. Since ϕ is an epimorphism, now k = 0, so K = 0. We omit the technical proof of the next lemma [Freyd, 2.64]. Lemma 2.29. Let 0 0 0

0 B11 B12 B13

0 B21 B22 B23

0 B31 B32

0 be a commutative diagram with exact columns and middle row in an abelian category. Then the top row is exact if and only if the bottom row is exact. Lemma 2.30 (The nine lemma). Let 0 0 0

ϕ1 ψ1 0 A1 B1 C1 0

f1 g1 h1

0 A2 B2 C2 0 ϕ2 ψ2

f2 g2 h2

0 A3 B3 C3 0 ϕ3 ψ3

0 0 0 be a commutative diagram with exact columns and the two bottom rows exact. Then the top row is exact. Proof. Apply the previous lemma and its dual.

25 3 Module categories

The goal of this section is to prove that the modules over a unitary ring give rise to an abelian category.

Deﬁnition 3.1. Let R be a unitary ring. A left R-module M is an abelian group M 0 together with an action R × M 0 → M 0, (r, m) 7→ rm, satisfying: (1) r(m + m0) = rm + rm0; (2) (r + r0)m = rm + r0m; (3) r0(rm) = (r0r)m for all m, m0 ∈ M 0 and r, r0 ∈ R. We identify M and M 0.

Deﬁnition 3.2. Let R be a unitary ring and M,N be R-modules, and let f : M → N be a group homomorphism. If f satisﬁes the additional property that f(rm) = rf(m), for all m ∈ M, r ∈ R, then f is a left R-map.

Theorem 3.3. For a ﬁxed ring R, the left R-modules and left R-maps give rise to a category.

Proof. We use the ordinary composition of functions. By the choice of composition, it is suﬃcient to show that the identity maps and any deﬁned com- positions of R-maps are R-maps. Suppose f : M1 → M2 and g : M2 → M3 are R-maps, and let r ∈ R, m ∈ M1. Now,

(g ◦ f)(rm) = g(f(rm)) = g(rf(m)) = rg(f(m)) = r(g ◦ f)(m), and

1M1 (rm) = rm = r1M1 (m), which prove the claim. We shall now consider a ﬁxed unitary ring R, and the left modules over this ring. Therefore, we shall refer to the left modules as modules, and to the R-maps as maps. Since R itself is an R-module, the category of left R-modules, denoted by R-mod, is always non-empty.

Deﬁnition 3.4. A submodule N of a module M is a subgroup of M such that RN ⊂ N.

Clearly, a submodule of a module is a module itself.

26 Lemma 3.5. For any modules M,N and any map f : M → N, the following are true: (1) M/M 0 is a module for any submodule M 0 of M; (2) ker f is a submodule of M; (3) im f is a submodule of N; (4) N/im f is a module.

Proof. (1) Let m, m0 ∈ M and r, r0 ∈ R. It is easy to check that the following hold: r((m + m0) + M 0) = r(m + M 0) + r(m0 + M 0), (r + r0)(m + M 0) = r(m + M 0) + r0(m + M 0), and r0(r(m + M 0)) = (r0r)(m + M 0) which prove the claim. (2) Because ker f is a subgroup of M, it is suﬃcient to show R(ker f) ⊂ ker f. Let m ∈ ker f and r ∈ R. Now f(rm) = rf(m) = r0 = 0, so rm ∈ ker f, proving the claim. (3) Since im f is a subgroup of N, it suﬃces to show R(im f) ⊂ im f. Let n ∈ im f and r ∈ R. Now, there is m ∈ M such that f(m) = n, so rn = rf(m) = f(rm) ∈ im f, proving the claim. (4) Immediate from (1) and (3).

Lemma 3.6. For any modules M and N, and any map f : M → N, the group theoretical kernel ker f is a category theoretical kernel of f.

Proof. Denote K = ker f, and let i: K → M be the inclusion map. Because kernels are unique up to isomorphism, it is sufficient to show there is a unique map h: X → K, for any module X and any map g : X → M such that f ◦ g = 0. Suppose X is a module, and g : X → M is such that f ◦ g = 0. Define h: X → K by x 7→ (i−1 ◦ g)(x). Now, h is well defined: if x ∈ X, then (f ◦ g)(x) = f(g(x)) = 0, so g(x) ∈ K, and so there is k ∈ K such that k = i(k) = g(x), and this k is unique because i is injective. Moreover, (i ◦ h)(x) = i(i−1(g(x))) = g(x), for every x ∈ X, so i ◦ h = g, and the uniqueness of h follows from the injectivity of i.

Lemma 3.7. For any modules M,N and any map f : M → N, the group theoretical cokernel coker f ∼= N/im f is a category theoretical cokernel of f. That is, the cokernels are modules.

27 Proof. Let X be any module, g : N → X be a map such that g ◦ f = 0, and let π : N → N/im f be the projection n 7→ n + im f. Now, π ◦ f = 0. Since cokernels are unique up to isomorphism, it is sufficient to show there is a unique h: N/im f → X such that h ◦ π = g. Define h by n + im f 7→ g(n). Now, h is well defined: if n + im f = n0 +im f, then h((n+im f)−(n0 +im f)) = h((n−n0)+im f) = g(n−n0) = g(n)−g(n0), but n−n0 ∈ im f, so g(n−n0) = 0 and h(n+im f) = h(n0+im f). Moreover, for any n ∈ N, (h ◦ π)(n) = h(n + im f) = g(n), so h ◦ π = g as required. The uniqueness of h follows from the surjectivity of π. We have now established that kernels and cokernels are modules.

Deﬁnition 3.8. Let J 6= ∅. The direct product of a family {Mj | j ∈ J} of Q modules is j∈J Mj with operations deﬁned coordinatewise: 0 0 (mj)j∈J + (mj)j∈J = (mj + mj)j∈J and r(mj)j∈J = (rmj)j∈J for all mj ∈ Mj and r ∈ R.

Definition 3.9. Let J 6= ∅. The direct sum of a family {Mj | j ∈ J} of P Q modules is j∈J Mj ⊂ j∈J Mj whose elements are of the form (mj)j∈J with mj 6= 0 for only a finite number of indices j. The operations are defined like those of the direct product. Lemma 3.10. (1) For any non-empty family of modules, the direct product and sum are modules. (2) The direct sum of a non-empty family of modules is a submodule of the direct product of the same family. (3) If a family of modules is finite, then its direct product and direct sum coincide. Proof. (1) It is easy to check the conditions are satisfied for both the sum and the product. (2) Since the direct sum is a subset of the direct product, and both are modules with the same operations, the direct sum is a submodule of the direct product. (3) If the family is finite, then the direct product is contained in the direct sum. Lemma 3.11. Let M,N be modules, and let f : M → N be a map. Then: (1) f is a monomorphism iff f is injective. (2) f is an epimorphism iff f is surjective.

28 Proof. (1) Suppose f is a monomorphism, and g : M → M is any map. If 0 0 now f ◦ g = f ◦ 1M , then g = 1M , and so f(m) = f(m ) implies m = m for all m, m0 ∈ M. Suppose then that f is injective, and let g, h: M 0 → M be any maps with f ◦ g = f ◦ h. Then f(g(m)) = (f ◦ g)(m) = (f ◦ h)(m) = f(h(m)) for all m ∈ M, but the injectivity of f implies g(m) = h(m) for every m ∈ M 0, and thus g = h and f is a monomorphism. (2) Suppose f is an epimorphism. Deﬁne g : N → N by (g|im f)(n) = n, for each n ∈ im f, and (g|(N \ im f))(n) = 0 otherwise. Now, g ◦ f = 1N ◦ f, but f is an epimorphism, so g = 1N . It follows that im f = N. Suppose ﬁnally that f is surjective. If now g ◦ f = h ◦ f, for some g, h: N → N 0, then g|im f = h|im f, but im f = N, so g = h and f is an epimorphism.

Lemma 3.12. (1) For any monomorphism f : M → N, there is a map g : N → N 0 such that im f is a kernel of g. (2) For any epimorphism f : M → N, there is a map g : M 0 → M such that im f is a cokernel of g.

Proof. (1) Choose N 0 = coker f, and let g : N → N 0 be the projection n 7→ n + im f. Now g(n) = 0 iﬀ n ∈ im f, so ker g = im f. The result now follows from 3.6. (2) Let M 0 = ker f, and let g : M 0 → M be the canonical inclusion. Now coker g = M/im g = M/M 0 ∼= im f = N since f is surjective, so im f is a cokernel of g. The result now follows from 3.7.

Lemma 3.13. The direct products and sums are products and coproducts, respectively, in R-mod.

Proof. We start by setting up some notation. Let {Mj | j ∈ J} be a family Q of modules, J 6= ∅, and let, for every k ∈ J, the map pk : Mj → Mk P j∈J be the projection (mj)j∈J 7→ mk, and λk : Mk → j∈J Mj be the injection mk 7→ (mj)j∈J with mj = 0 for all j 6= k. Let then N be any module, and, for each j ∈ J, let fj : N → Mj be any Q map. To show that Mj is a categorical product, we need to show there Qj∈J is a unique h: N → j∈J Mj such that fj = pj ◦ h, for all j ∈ J. Deﬁne h(n) = (fj(n))j∈J , so that now pj ◦ h = fj for each j ∈ J. Moreover, if 0 Q 0 0 h : N → j∈J Mj satisﬁes pj ◦ h = fj for all j ∈ J, then pj ◦ (h − h ) = 0 for 0 T T all j ∈ J. Thus, we obtain im (h − h ) ⊂ j∈J ker pj, but j∈J ker pj = 0. It follows that im (h − h0) = 0, so h = h0, proving the uniqueness of h. P Let then gj : Mj → N be any map for each j ∈ J. To show Mj is P j∈J a categorical coproduct, we must show there is a unique µ: j∈J Mj → N

29 P P such that µ ◦ λj = gj, for all j ∈ J. Now, define µ( j∈J mj) = j∈J gj(mj), so that (µ ◦ λj)(mj) = µ(λj(mj)) = µ(mj) = gj(mj), for each j ∈ J. Now, if P ν : j∈J Mj → N satisfies ν ◦ λj = gj for all j ∈ J, then X X (µ − ν)( mj) = (µ(mj) − ν(mj)) j∈J j∈J X = (gj(mj) − gj(mj)) j∈J = 0, P P for any j∈J mj ∈ j∈J Mj, so µ − ν = 0, proving the uniqueness of µ. Lemma 3.14. The trivial module is a zero object. Proof. Let M and N be any modules, and let f : M → N be such that f(m) = 0 for all m ∈ M. Now, f(m + m0) = 0 = 0 + 0 = f(m) + f(m0), and f(rm) = 0 = r0 = rf(m), for all m, m ∈ M, r ∈ R, so f is an R-morphism. Now, if M = 0, then f is the unique morphism 0 → N, and similarly, if N = 0, then f is the unique morphism M → 0, so 0 is both an initial and a terminal object. Combining these results, we have now shown the theorem: Theorem 3.15. For any fixed unitary ring R, R-mod is an abelian category.

In particular, since abelian groups are R-modules with R = Z, we conclude that Ab is abelian.

4 On functors

In this section we will consider properties of functors, and look at the connec- tion between certain types of abelian categories and the categories of modules over unitary rings. Deﬁnition 4.1. A covariant functor F : C → D is a morphism of categories such that: (1) if A ∈ obj C, then F (A) ∈ obj D, (2) if f ∈ Hom(A, B), where A, B ∈ obj C, then F (f) ∈ Hom(F (A),F (B)), (3) if g ◦ f is deﬁned, then F (g ◦ f) = F (g) ◦ F (f), (4) for all objects A, F (1A) = 1F (A). A contravariant functor F : C → D is a covariant functor F : Cop → D. Covariant functors shall be called functors for the remaining text.

30 Deﬁnition 4.2. A category is called locally small if the classes Hom(A, B) are sets for all objects A, B of the category.

An example of a locally small category is the category of sets. It follows that every category in which the morphisms are some kind of functions, that is, for example continuous or some algebraic structure preserving, will be locally small as well, since their morphisms can be identiﬁed with functions between the underlying sets. In particular, the categories of topological spaces, of all groups, of rings, and of modules over any ring are locally small. It is also obviously the case that if a category is locally small, then so is its opposite category.

Deﬁnition 4.3. Let C, D be locally small categories, and let F : C → D be a functor. For every A, B ∈ obj C, F induces a function FAB : Hom(A, B) → Hom(F (A),F (B)), f 7→ F (f). (1) F is called faithful if FAB is injective for all objects A, B, (2) F is called full if FAB is surjective for all objects A, B, (3) F is called fully faithful if it is full and faithful.

Deﬁnition 4.4. A functor F : C → D is injective if F (A) = F (B) implies A = B for all objects A, B of C.

Deﬁnition 4.5. A functor is an embedding if it is injective and fully faithful.

Deﬁnition 4.6. For any locally small category C and an object A of C, Hom(A, ·) and Hom(·,A) are the covariant and contravariant representable functors, respectively, C → Sets deﬁned by (1) if g : B → C, then Hom(A, g):(f : A → B) 7→ (g ◦ f : A → C), (2) if h: C → B, then Hom(h, A):(f : B → A) 7→ (f ◦ h: C → A) for any objects B,C of C.

Deﬁnition 4.7. Let C, D be categories and F,G: C → D be functors. A natural transformation τ : F → G is a class of morphisms {τA | A ∈ obj C} such that the diagram

F (f) F (A) F (B)

τA τB G(A) G(B) G(f) commutes for all A, B ∈ obj C and all f : A → B. A natural transformation τ is a natural equivalence if τA is an isomorphism for every object A.

31 The deﬁnition for contravariant functors is the same with the horizontal arrows reversed. Denote the class of natural transformations from a functor F to a functor G by Nat(F,G).

Deﬁnition 4.8. For any locally small category C, deﬁne SetsC to be the category with functors C → Sets of either variance as objects and natural transformations as morphisms. Then SetsC is said to be a set-valued functor category.

Theorem 4.9 (The Yoneda Lemma). Let C be a locally small category and op let F : Cop → Sets be a covariant functor. Deﬁne y : C → SetsC , C 7→ Hom(·,C). Then there is a bijection Nat(y(C),F ) → F (C), natural in C and F .

Proof. Let C ∈ obj C be arbitrary. Deﬁne a function η = ηC,F : Nat(y(C),F ) → F (C) by η(θ) = θC (1C ) for all θ ∈ Nat(y(C),F ), and denote xθ = θC (1C ). Conversely, for any a ∈ F (C), deﬁne a natural transformation θa : y(C) → F 0 0 0 by setting (θa)C0 : Hom(C ,C) → F (C ), h 7→ F (h)(a) for all h: C → C. To 0 00 see that θa is indeed a natural transformation, let f : C → C be arbitrary, and consider the diagram

f ∗ Hom(C00,C) Hom(C0,C)

(θa)C00 (θa)C0 F (C00) F (C0), F (f)

∗ ∗ where f = Hom(f, C). Now, ((θa)C0 ◦f )(h) = (θa)C0 (h◦f) = F (h◦f)(a) = 0 00 (F (f) ◦ F (h))(a) = (F ◦ (θa)C00 ))(h) for each h: C → C , so the diagram commutes. Therefore θa is a natural transformation y(C) → F .

Next, let θ : y(C) → F be arbitrary, and consider θxθ . By deﬁnition, 0 0 for any h: C → C, we have (θxθ )C (h) = F (h)(xθ) = F (h)(θC (1C )). The naturality of θ makes the square

y(C)(h) y(C)(C) y(C)(C0)

θC θC0 F (C) F (C0) F (h) commute, and so F (h)(θC (1C )) = (F (h) ◦ θC )(1C ) = (θC0 ◦ y(C)(h))(1C ) = 0 θC (h), so it follows that θxθ = θ. On the other hand, xθa = (θa)C (1C ) = F (1C )(a) = 1FC (a) = a, so deﬁning ψ = ψC,F : F (C) → Nat(y(C),F ) by ∼ ψ(a) = θa, one obtains an inverse for η. Therefore, we have Nat(y(C),F ) = F (C).

32 Let then ϕ: F → F 0 be a natural transformation, and consider the diagram η Hom(y(C),F ) C,F F (C)

ϕ∗ ϕC Hom(y(C),F 0) F 0(C), ηC,F 0 where ϕ∗ = Hom(y(C), ϕ). For every θ ∈ Nat(y(C),F ), we have

ϕC (xθ) = ϕC (θC (1C ))

= (ϕ ◦ θ)C (1C )

= xϕθ

= ηC,F 0 (ϕ ◦ θ)

= (ηC,F 0 (Hom(y(C), ϕ)(θ)), proving the commutativity. Thus, the isomorphism is natural in F . Finally, let f : C0 → C be a morphism in C. For every θ ∈ Nat(y(C),F ), we have

(ηC0,F ◦ Hom(y(C)(f),F ))(θ) = ηC0,F (θ ◦ y(C)(f))

= (θ ◦ y(C)(f))C0 (1C0 )

= (θC0 ◦ y(C)(f))C0 )(1C0 )

= θC0 (f ◦ 1C0 )

= θC0 (f)

= θC0 (1C ◦ f)

= (θC0 ◦ y(C)(f))(1C )

= (F (f) ◦ θC )(1C ), where the last equation holds because the square

y(C)(f) y(C)(C) y(C)(C0)

θC θC0 F (C) F (C0) F (f) commutes. On the other hand, (F (f) ◦ θC )(1C ) = (F (f) ◦ ηC,F )(θ), so the isomorphism is natural in C.

Corollary 4.9.1. Given objects A, B in any locally small category, y(A) ∼= y(B) implies A ∼= B.

33 Proof. Suppose there exists an isomorphism ϕ: y(A) → y(B), for some objects A, B. Since the functor y is full, there is now a morphism θ : A → B such that ϕ = y(θ), and since y is faithful, θ is an isomorphism. Definition 4.10. A locally small category is small if its class of objects is a set. The category of sets is locally small, but not small, as otherwise its set of objects would contain itself as an element, which contradicts the assumption that the class of sets was a set itself. On the other hand, given any set X, one can construct a category by taking the power set P(X) using its elements as objects and their identity functions as morphisms, or one can consider the empty category with no objects or morphisms. These are small categories, so the definition makes sense. Lemma 4.11. If A, B, C ∈ obj C and m: B → C is a monomorphism, then Hom(A, m) is a monomorphism. Proof. Let f, g ∈ Hom(A, B). Suppose Hom(A, m)(f) = Hom(A, m)(g). Then m ◦ f = m ◦ g, so f = g and Hom(A, m) is a monomorphism. Definition 4.12. Let 0 A B C 0 be a short exact sequence. A functor F is left exact if: (1) F is covariant and

F (0) F (A) F (B) F (C) is exact or (2) F is contravariant and

F (C) F (B) F (A) F (0) is exact, and F is right exact if: (1) F is covariant and

F (A) F (B) F (C) F (0) is exact or (2) F is contravariant and

34 F (0) F (C) F (B) F (A) is exact. F is exact if it is both left and right exact.

Deﬁnition 4.13. Let F : A → B be a functor between abelian categories. The functor F is additive if, for any A, B ∈ obj A and any f, g ∈ Hom(A, B), F (f + g) = F (f) + F (g).

Consider the nth homology functor Hn : CompAb → Ab given by Hn(A∗) = ker dn/im dn+1. One can show that Hn(f∗ + g∗) = Hn(f∗) + Hn(g∗) for any f∗, g∗ : A∗ → B∗. Lemma 4.14. Let F : A → B be an additive functor between abelian categories. If 0 is the zero object of A, then F (0) is the zero object of B.

Proof. Let f be a zero morphism in A. Then F (f) = F (f+f) = F (f)+F (f), and so F (f) = 0. Now, since F (10) = 1F (0) is the identity of F (0), and 10 is a zero morphism, 1F (0) is a zero morphism, and F (0) is a zero object. Theorem 4.15. Let A be a locally small abelian category, and let A be any object of A. Then Hom(A, ·) is additive and left exact.

Proof. Suppose B,C ∈ obj A, and let f, g ∈ Hom(B,C) be arbitrary. For any ϕ ∈ Hom(A, B), we have Hom(A, f +g)(ϕ) = (f +g)◦ϕ = f ◦ϕ+g ◦ϕ = Hom(A, f)(ϕ) + Hom(A, g)(ϕ). Let then f g 0 B0 B B00 0 be a short exact sequence. Since Hom(A, ·) is additive, it suﬃces to show that Hom(A, f) is a monomorphism and that im Hom(A, f) = ker Hom(A, g). Let ϕ, θ ∈ Hom(A, B0), and suppose Hom(A, f)(ϕ) = Hom(A, f)(θ). Now f ◦ϕ = f ◦θ, but f is a monomorphism by exactness, so ϕ = θ. Since g◦f = 0, Hom(A, g) ◦ Hom(A, f) = Hom(A, g ◦ f) = 0 by additivity. Therefore it is suﬃcient to show ker Hom(A, g) ⊂ im Hom(A, f). If θ ∈ ker Hom(A, g), then g ◦ θ = 0. Since ker g = im f, there is a u: A → B0 such that θ = f ◦ u = Hom(A, f)(u), and so ker Hom(A, g) ⊂ im Hom(A, f). Similarly, Hom(·,B) is left exact for all objects B in an abelian category.

Deﬁnition 4.16. An object G is a generator for an abelian category A if the functor Hom(G, ·): A → Ab is faithful.

35 Lemma 4.17. Let A be an abelian category, and let G ∈ obj A. Then the following conditions are equivalent: (1) G is a generator for A; (2) for any morphisms f, g : A → B in A such that f 6= g, there is some morphism h: G → A such that f ◦ h 6= g ◦ h. Proof. Suppose (1) holds, and let f, g : A → B be arbitrary morphisms in A such that f 6= g. Now, since Hom(G, ·) is faithful, f∗ = Hom(G, f) 6= Hom(G, g) = g∗, so there is some h ∈ Hom(G, A) such that f ◦ h = f∗(h) 6= g∗(h) = g ◦ h. Conversely, suppose (2) holds. Let f, g : A → B, f 6= g, be arbitrary morphisms in A. By assumption, there is a morphism h: G → A such that f∗(h) = f ◦ h 6= g ◦ h = g∗(h), so f∗ 6= g∗, and the functor is faithful. Deﬁnition 4.18. An object C is a cogenerator for an abelian category A if the functor Hom(·,C): A → Ab is faithful. Lemma 4.19. Let A be an abelian category, and let C ∈ obj A. Then the following conditions are equivalent: (1) C is a cogenerator for A; (2) for any morphisms f, g : A → B in A such that f 6= g, there is some morphism h: B → C such that h ◦ f 6= h ◦ g. Proof. Similar to the proof of the previous lemma. Deﬁnition 4.20. An object P in an abelian category A is projective if the functor Hom(P, ·): A → Ab is exact. Lemma 4.21. Let A be an abelian category, and let P ∈ obj A. Then the following conditions are equivalent: (1) P is projective; (2) for any epimorphism p: A → B and any f : P → B in A, there is a morphism g making the diagram

P g f

A p B 0 commutative. Proof. Suppose (1) holds. Let p: A → B be an epimorphism in A, and let f : P → B be arbitrary. If k : K → A is a kernel of p, then the sequence p 0 K k A B 0

36 is exact, so we obtain, by assumption, an exact sequence

p 0 Hom(P,K) k∗ Hom(P,A) ∗ Hom(P,B) 0 of abelian groups. By exactness, p∗ is surjective, so there is some g ∈ Hom(P,A) such that f = p∗(g) = p ◦ g. Conversely, suppose (2) holds. Let p 0 K k A B 0 be an exact sequence in A. By 4.15, the sequence

p 0 Hom(P,K) k∗ Hom(P,A) ∗ Hom(P,B) is exact, so it is suﬃcient to show that p∗ is surjective. Let f : Hom(P,B) be arbitrary. By assumption, there is a morphism g : P → A such that p◦g = f, but then p∗(g) = f, and thus p∗ is surjective and the functor is exact. Free abelian groups provide an example of projective objects. In order to see this, let F be a free abelian group, p: A → B be a surjective homomorphism between any abelian groups, and let f : F → B be arbitrary. If X ⊂ F is a basis for F , we can choose for each x ∈ X an ax ∈ A such that p(ax) = f(x). Now the universal property of free abelian groups gives the existence of a homomorphism g : P → A such that g(x) = ax for each x ∈ X, so, for any y ∈ F , we have

n X (p ◦ g)(y) = (p ◦ g)( mkxk) k=1 n X = (mk(p ◦ g)(xk)) k=1 n X = (mkf(xk)) k=1 n X = f( mkxk) k=1 = f(y).

It follows that p ◦ g = f.

Deﬁnition 4.22. An object E in an abelian category A is injective if the functor Hom(·,E): A → Ab is exact.

37 Lemma 4.23. Let A be an abelian category, and let E ∈ obj A. Then the following conditions are equivalent: (1) E is injective; (2) for any monomorphism m: A → B and any f : A → E in A, there is a morphism g making the diagram

E g f

0 A m B commutative.

Proof. Similar to the proof of the previous lemma. It follows that objects E,P are an injective and a projective object, respectively, if and only if Hom(·,E) and Hom(P, ·) are right exact. Also, by noticing that the conditions 4.17(2) and 4.21(2) are dual to the conditions 4.19(2) and 4.23(2), respectively, we obtain Lemma 4.24.

Lemma 4.24. Generators are dual to cogenerators, and projectivity is dual to injectivity.

Deﬁnition 4.25. An abelian subcategory A0 of an abelian category A is exact if the inclusion functor A0 → A is exact.

Lemma 4.26. For every set {Ai}i∈I of objects in an abelian category A, there 0 0 is an abelian, full, small and exact subcategory A of A such that Ai ∈ obj A for all i ∈ I.

Proof. Let K,F : Hom A → obj A and S : obj A × obj A → obj A be defined by K(f) = ker f, F (f) = coker f and S(A, B) = A ⊕ B. Given a non-empty subcategory B of A, define C(B) to be the full subcategory generated by B, K(B), F (B) and S(B × B), that is, a full subcategory of A such that the class of objects in B, K(B), F (B) and S(B × B) is a subclass of the class of objects in C(B). If B is small, then so is C(B): If B is small, obj B, K(B), F (B) and S(B × B) are sets, so their union is a set, but this is the class of objects in C(B), so C(B) is small. ∞ Defining Cn+1(B) = C(Cn(B)), we claim that C∞(B) = S Cn(B) is a n=1 full and exact abelian subcategory of A. (1) Since obj B 6= ∅, there is A ∈ obj B, and thus 1A ∈ HomB(A, A). ∞ Since K(1A) = 0, the zero object is in C (B).

38 (2) If A, B ∈ obj B, then S(A, B) is a C(B)-direct sum, but the fullness of C(B) makes S(A, B) an A-sum, and so S(A, B) is a C∞(B)-sum. (3) If A, B ∈ obj B then HomB(A, B) = HomA(A, B), and so ker f is in C∞(B). (4) If A, B are objects in B, then HomB(A, B) = HomA(A, B), and ∞ so f ∈ HomB(A, B) is an A-monomorphism if and only if it is a C (B)- monomorphism. Suppose f is an A-monomorphism. Then, take any C ∈ obj C∞(B) and g : B → C, such that

f g 0 A B C 0 is exact in A. Now, f is an A-kernel, and so a C∞(B)-kernel, of g. Finally, if

f g 0 A B C 0 is a short exact sequence in C∞(B), then it is a short exact sequence in A, by fullness. Choosing now obj B = {Ai}i∈I , one obtains the claim. Deﬁnition 4.27. A diagram scheme (or a scheme) is a small category, and a diagram in a category A is a functor from a diagram scheme into A. A set of exactness conditions on a scheme is a set of ordered pairs of morphisms in the scheme. Given a scheme S, a set E of exactness conditions, and a diagram D : S → A, we say D satisﬁes the exactness conditions if, for every pair (x, y) ∈ E, (D(x),D(y)) is an exact sequence in A.

For example, if A, B, C are objects in S, and x: A → B, y : B → C are such that (x, y) ∈ E, then we are interested in if

D(x) D(y) D(A) D(B) D(C) is exact. If it is, we say (D(x),D(y)) is an exact sequence (in A).

Deﬁnition 4.28. A simple diagrammatic statement is a statement about the exactness and commutativity of a diagram. A compound diagrammatic statement is of the form P → Q, where P,Q are simple diagrammatic state- ments.

Deﬁnition 4.29. An abelian category A is very abelian if there is an exact embedding A0 → Ab for every small, exact and abelian subcategory A0 of A.

Theorem 4.30 (The First Metatheorem). Every compound diagrammatic statement true in Ab is true in every very abelian category.

39 Proof. Suppose (S,E1,E2) is true in Ab. Let D : S → A be a diagram in 0 a very abelian category A satisfying the exactness conditions E1. Let A be a small and exact abelian subcategory of A such that D(S) lies in A0 and 0 0 D satisfies E2 in A if and only if it satisfies E2 in A . Let F : A → Ab be an exact embedding. Now F ◦ D : S → Ab satisfies E1, and it satisfies E2 if 0 and only if D : S → A satisfies E2. Definition 4.31. A map extension of a scheme S is a scheme S¯ together with a functor G: S → S¯ such that G gives a bijection obj S → obj S¯. Given a scheme S, a map extension S → S¯, and sets E, E¯ of exactness conditions for S, S¯, respectively, we say that the full compound diagrammatic statement (S → S,E,¯ E¯) is true for A if, for every diagram D : S → A satisfying E, there is a diagram D¯ : S¯ → A satisfying E¯ and D = D¯ ◦ G. Definition 4.32. An abelian category A is fully abelian if for every full, small and exact subcategory A0 there is a ring R and a full exact embedding A0 → R-mod. Theorem 4.33 (The full metatheorem). If a full compound diagrammatic statement is true for all categories of R-modules, then it is true for all fully abelian categories. Proof. The proof is similar to that of the first metatheorem. Definition 4.34. A category A is left-complete if every pair of morphisms has a difference kernel and every indexed set of objects has a product, and right-complete if every pair of morphisms has a difference cokernel and every indexed set of morphisms has a sum. If A is both left- and right-complete, then it is complete. The category Ab is left-complete. This can be seen by noticing that Q A = i∈J Aj is an abelian group if every Aj is, and that kernels of f − g are difference kernels of f and g for any morphisms f, g : A → B in Ab.A similar consideration shows that Ab is also right-complete. Theorem 4.35. A complete abelian category with a projective generator is fully abelian. Proof. Let A0 be a small, full and exact subcategory of a complete abelian category A, and P¯ a projective generator for A. For each A ∈ obj A0, ˜ P ¯ ¯ ¯ consider the object P = Pf , where each Pf = P . For each f ∈ f∈Hom(P¯ ,A) ¯ ¯ ˜ Hom(P,A), let ϕf : Pf → A be f, and let ϕ: P → A be the unique morphism ¯ such that ϕ ◦ λf = ϕf = f for every f ∈ Hom(P,A).

40 We claim ϕ is an epimorphism. Indeed, if p, q : A → B are any morphisms such that p ◦ ϕ = q ◦ ϕ, then

p ◦ f = p ◦ (ϕ ◦ λf )

= (p ◦ ϕ) ◦ λf

= (q ◦ ϕ) ◦ λf

= q ◦ (ϕ ◦ λf ) = q ◦ f, for every f ∈ Hom(P,A¯ ), so p = q by Lemma 4.17 (2). Taking I = S ¯ P ¯ Hom(P,A) and deﬁning P = I P , one obtains a projective gen- A∈obj A0 erator P such that there is an epimorphism P → A, for every A ∈ obj A0. Choose R = Hom(P,P ) to get a unitary ring. For every A ∈ obj A, the abelian group Hom(P,A) has a canonical R-module structure: if x: P → A, r : P → P , then rx = x ◦ r ∈ Hom(P,A). Given a morphism f : A → B with A, B ∈ obj A, the induced morphism f¯: Hom(P,A) → Hom(P,B) is an R-morphism, since

f¯(rx) = f ◦ (x ◦ r) = (f ◦ x) ◦ r = rf¯(x).

We can, therefore, define F : A → R-mod by F (A) = Hom(P,A) with the canonical R-module structure. Since P is a projective generator, F is an exact embedding. It suffices to show F |A0 is full. Given any A, B ∈ obj A0 and a morphism f¯: F (A) → F (B), we must find f : A → B such that F (f) = f¯. Let 0 K P A 0 and P B 0 be exact sequences in A. Since F (P ) = R, we obtain the commutative diagram 0 F (K) R F (A) 0

g f¯ R F (B) 0

41 in R-mod, where the existence of g follows from the projectiveness of R in R-mod. Since R is a ring, g must be of the form g(s) = sr for all s ∈ R and some r ∈ R. Returning to A, the diagram p 0 K α P A 0 r P B 0 β is such that β ◦ r ◦ α = 0, since F is an embedding and F (β) ◦ g ◦ F (α) = 0. Since the top row is exact, p: P → A is a cokernel of α, and so there is a morphism f : A → B such that the diagram p P A

r f P B β commutes. Hence R F (A)

g F (f) R F (B) commutes. Since R → F (A) is epimorphic, now F (f) = f¯.

5 On the Grothendieck property and injectivity

In this section we will prove results about injective objects and consider categories satisfying the distributivity condition of unions and intersections. After Deﬁnition 5.3., we will assume every category in this section to be a Grothendieck category unless stated otherwise.

Deﬁnition 5.1. Given an object A in an abelian category, an extension of A is a monomorphism i: A → B. This extension is trivial if it splits, that is, there is a morphism q : B → A such that q ◦ i = 1A. Lemma 5.2. An object E in an abelian category A is injective if and only if it only has trivial extensions.

42 Proof. Suppose E is injective. An object P is projective if and only if for every epimorphism f : A → A00 and any morphism g : P → A00 there is a morphism h: P → A such that f ◦h = g. Dually, E is injective if and only if, for every monomorphism f : A00 → A and any morphism g : A00 → E, there is a morphism h: A → E such that h ◦ f = g. Now, choose A00 = E and g = 1E to get the split h ◦ f = 1E. Conversely, suppose E has only trivial extensions. Let f : A → B be a monomorphism and g : A → E arbitrary. Consider the pushout diagram

f A B g ϕ p E P. By the pushout theorem, p is a monomorphism, so p is a trivial extension of E, by hypothesis. Let q : P → E be such that q ◦ p = 1E, and deﬁne θ : B → E by θ = q ◦ ϕ. Now the diagram

f A B

g θ E commutes and E is injective. Recall that the subobjects are ordered naturally by containment as in Deﬁnition 1.17. Therefore, the following deﬁnition makes sense.

Deﬁnition 5.3. A Grothendieck category is a complete and well-powered abelian category A satisfying the condition (AB5) If {Ai}i∈I is a linearly ordered family of subobjects and B is any S S object in A, then B ∩ Ai = (B ∩ Ai). i∈I i∈I For example, the categories Grp and Top satisfy this condition which they inherit from the facts that their objects are based upon sets, and that the category Sets satisﬁes the condition. Moreover, they are all well-powered, but none of them is abelian. On the other hand, the category Ab in- herits well-poweredness and the property (AB5) from Grp, and is thus a Grothendieck category.

Deﬁnition 5.4. Let then C, D be any categories, and let F,G: C → D be functors. We say G is a subfunctor of F if there is a natural transformation τ : G → F such that τA : G(A) → F (A) is a monomorphism for every A ∈ obj C. We denote this by G ⊂ F .

43 Lemma 5.5. Let A be any small abelian category. The category AbA of additive functors A → Ab is a Grothendieck category.

Proof. Given a family {Fi}i∈I of subfunctors of a functor F , construct their union and intersection pointwise: S S T T ( Fi)(A) = (Fi(A)) ⊂ F (A) and ( Fi)(A) = (Fi(A)).

Now, given a linearly ordered family {Fi}i∈I and a subfunctor H ⊂ F , we S S S S have (H ∩ Fi)(A) = H(A)∩ Fi(A) = [H(A)∩Fi(A)] = [ (H ∩Fi)](A).

Deﬁnition 5.6. An essential extension of an object A in an abelian category is a monomorphism i: A → B such that, for every non-zero monomorphism f : B0 → B, im i ∩ im f 6= 0. We also call an extension i: A → B proper if im i ∩ B 6= B.

Lemma 5.7. In a Grothendieck category A, if E is an object with no proper essential extensions and i: E → B is an extension of E in A and F is the partially ordered family of subobjects of B such that B0 ∩ im i = 0 for all 0 S B ∈ F, then Bi ∈ F, for any ascending chain {Bi}i∈I in F.

Proof. Let A = im i. Now, for any ascending chain {Bi}i∈I in F, we have 0 S S B = A ∩ Bi = (A ∩ Bi). Since all of the intersections are trivial, it follows that B0 = 0.

Theorem 5.8. In a Grothendieck category A, an object is injective if and only if it has no proper essential extensions.

Proof. If E is injective, its only proper extensions are trivial, and thus not essential. Suppose E has no proper essential extensions, and consider an extension i: E → B. We show that this extension is trivial. Let F be the partially ordered family of subobjects of B with trivial intersections with im i. By the preceeding lemma, all ascending chains have upper bounds in F, so Zorn’s lemma applies and there is a maximal object B0 in F. Let F ∗ be the corresponding family of quotient objects of B, that is, F ∈ F ∗ if and only if there is a morphism p: B → F such that p ◦ i is a monomorphism. Then F ∗ has a minimal element B∗, and there is a morphism q : B → B∗ such that q ◦ i is a monomorphism. Moreover, since B∗ is minimal, q ◦ i is essential. Indeed, if f : B∗ → F is such that f ◦ q ◦ i is monomorphic, then coim q ∈ F ∗ is a subobject of B∗, and minimality gives coim q = B∗. By hypothesis, q ◦ i is not a proper essential extension, so it is an isomorphism E ∼= B∗. Let g : B∗ → E be its inverse. Now (g ◦ q) ◦ i = 1E is a split.

44 Deﬁnition 5.9. An injective envelope of an object A is an injective essential extension A → E, where E is injective.

Lemma 5.10. An essential extension of an essential extension is essential.

Proof. Immediate from the deﬁnition.

Lemma 5.11. Let e: A → E be an extension of A in a Grothendieck category, and {Ei}i∈I an ascending chain of subobjects between im e and E. If S Ei is an essential extension of A for each i ∈ I, then so is Ei. In other i∈I words, every ascending chain of essential extensions has an upper bound that is, also, an essential extension. S S Proof. Let S be an arbitrary non-trivial subobject of Ei. Then S∩ Ei = i∈I i∈I S (S ∩ Ei), and S ∩ Ej 6= 0 for some j ∈ I. Since Ej is an essential extension i∈I of A, now S ∩ A 6= 0.

Theorem 5.12. Let B be a Grothendieck category, I an ordered set, and j {ϕi : Ei → Ej}i,j∈I,i≤ja family of monomorphisms such that, for i ≤ j ≤ k, the diagram

k ϕi Ei Ek

j k ϕi ϕj Ej commutes. Then there is an object E and a family {ϕi : Ei → E}i∈I of monomorphisms in B such that, for i ≤ j, the diagram

ϕi Ei E

j ϕj ϕi Ej commutes. P Proof. Let S = Ei, and, for each i ∈ I, let λi : Ei → S be the associated i∈I morphism. For each j ∈ I, deﬁne hj : S → S to be the unique morphism such that ( j λj ◦ ϕi , if i ≤ j hj ◦ λi = λi, if j ≤ i.

45 0 If j ≤ j , then hj0 = hj0 ◦ hj. Indeed, if i ≤ j, then, by uniqueness,

(hj0 ◦ hj) ◦ λi = hj0 ◦ (hj ◦ λi) j = hj0 ◦ (λj ◦ ϕi ) j = (hj0 ◦ λj) ◦ ϕi j0 j = (λj0 ◦ ϕj ) ◦ ϕi j0 j = λj0 ◦ (ϕj ◦ ϕi ) j0 = λj0 ◦ ϕi

= hj0 ◦ λi, and otherwise (hj0 ◦ hj) ◦ λi = hj0 ◦ (hj ◦ λi) = hj0 ◦ λi. Therefore, {ker hi}i∈I is an ascending chain. S Let E = im hi, and deﬁne H := {hi : S → S | i ∈ I}. We order H by i∈I hi ≤ hj if hj = hj ◦ hi. It follows that this ordering coincides with that of I, and so there is an upper bound for any ascending chain in H. By Zorn’s lemma, there is now an index m ∈ I such that hm = hm ◦hj for all j ∈ I, and so E = im hm. Since every morphism in an abelian category has an image, we obtain a factorization S hm S

h α E, where h: S → E is an epimorphism and α: E → S is a monomorphism. S Since α is a monomorphism, ker hm = ker h, and ker hm = ker hi. We i∈I set ϕi = h ◦ λi. Now ϕi is a monomorphism if ker ϕi = ker λi = 0, that is, ker h ∩ im λi = 0. By the Grothendieck property, we have ker h ∩ im λi = S S ( ker hj)∩im λi = (ker hj ∩im λi), and, for each i ∈ I, we have ker hj ∩ j∈I j∈I im λi = 0, by construction.

46 Finally, let i ≤ j. We have

j j α ◦ (ϕj ◦ ϕi ) = α ◦ ((h ◦ λj) ◦ ϕi ) j = α ◦ (h ◦ (λj ◦ ϕi ))

= α ◦ (h ◦ (hj ◦ λi))

= (α ◦ h) ◦ (hj ◦ λi)

= hm ◦ (hj ◦ λi)

= (hm ◦ hj) ◦ λi

= hm ◦ λi

= (hm ◦ hi) ◦ λi

= hm ◦ (hi ◦ λi)

= hm ◦ λi

= (α ◦ h) ◦ λi

= α ◦ (h ◦ λi)

= α ◦ ϕi.

j Since α is a monomorphism, it follows that ϕj ◦ ϕi = ϕi, as desired. Let f : A → B be a morphism in a category with pullbacks. We recall that one can deﬁne f −1B to be the subobject of A such that the square

f −1B i A

f 0 f im f B, j where i, j are monomorphisms and f 0 is induced by f, is a pullback, and that then f −1B is unique up to isomorphism. One can easily check that this agrees with the way preimages are deﬁned for sets and functions.

Theorem 5.13. In a Grothendieck category with a generator, every object has an injective envelope.

Proof. Let B be a Grothendieck category. Using the axiom of choice, let E : obj B → Hom B be such that E(A) = iA : A → B, where iA is a monomorphism and B is a proper essential extension of A, unless A is injective, in which case B = A. γ γ+1 γ+1 For each ordinal number γ, deﬁne E (A) by E (A) = iA : E(A) → Eγ(A) → E(Eγ(A)), and, for a limit ordinal α, we let Eα(A) be a minimal essential extension for all Eγ(A), γ < α. Such an object exists due to the

47 previous theorem. Now, the ascending chain {Eγ(A)} becomes stationary only when it reaches an injective envelope of A. Suppose B has a generator G, and that {Eγ} is a sequence of essential extensions throughout the entire sequence of ordinal numbers. The goal is to show this eventually becomes stationary. For any monomorphism G0 → G and ordinal number γ, the family 0 {im [Hom(G, Eα) → Hom(G ,Eγ)]}α>γ is an ascending chain of subsets of 0 0 Hom(G ,Eγ). Since there is only a set of subsets of Hom(G ,Eγ), the chain must stabilize, and since there is only a set of subobjects of G, it follows that there is an ordinal number F (γ) such that im [Hom(G, Eα) → 0 0 Hom(G ,Eγ)] ⊂ im [Hom(G, EF (γ)) → Hom(G ,Eγ)] for all α < γ and all G0 ⊂ G. Since it is sufficient to show that any cofinal subsequence of {Eγ} stabi- lizes evetually, we may suppose F (γ) = γ + 1. Now, let Ω be the first ordinal of cardinality greater than that of the family of subobjects of G. We claim EΩ+1 = EΩ. Supposing otherwise, let f : G → E be a morphism such that im f 6⊂ im (EΩ → EΩ+1). For all γ < Ω + 1, make the identification Eγ = im (Eγ → −1 EΩ+1). The family {f Eγ} of subobjects of G is an ascending family. By the choice of Ω, it must stabilize before Ω. −1 −1 There exists, then, an ordinal γ < Ω such that f Eγ = f EΩ. By our assumption that F (γ) = γ + 1, we obtain a morphism g : G → EΩ+1 such −1 that im g ⊂ EΩ ⊂ EΩ+1, f 6= g, and ker (f − g) = f EΩ. Since f 6= g, im (f − g) 6= 0. If it was the case that EΩ ∩ im (f − g) 6= 0, then there would be a morphism h: G → G such that 0 6= im ((f − g) ◦ h) ⊂ −1 EΩ, so that im h ⊂ f EΩ, so (f − g) ◦ h = 0, which is a contradiction. Therefore, EΩ ∩ im (f − g) = 0. Lemma 5.14. If an object E in AbA is injective, then it is a right-exact functor.

Proof. Let A0 A A00 0 be any exact sequence in A. Applying the representable functor, we obtain the exact sequence 0 Hom(A00, ·) Hom(A, ·) Hom(A0, ·). The functor Hom(·,E) is exact, so we obtain a commutative diagram

48 Hom(Hom(A0, ·),E) Hom(Hom(A, ·),E) Hom(Hom(A00, ·),E) 0

E(A0) E(A) E(A00) 0, where the vertical arrows are the isomorphisms given by the Yoneda lemma.

6 Two embedding theorems

In this section we will prove the embedding theorem into the category of abelian groups, and then construct the remaining theory needed to prove the embedding theorem into the category of modules over a unitary ring. Definition 6.1. A mono functor is a functor that preserves monomorphisms. Definition 6.2. Let F,G: C → D be functors. An embedding ι: F → G is a natural transformation such that ιC : F (C) → G(C) is a monomorphism for all objects C in C. Lemma 6.3. Let A be a small abelian category. Then the essential extensions of mono functors A → Ab are mono functors. Proof. Let M be a mono functor, and let E be an essential extension of M in AbA. Suppose E is not a mono functor. Then there is a monomorphism f : A0 → A in A such that E(f) is not a monomorphism in Ab. Let 0 6= x ∈ E(A0) be such that E(f)(x) = 0. We construct the subfunctor F ⊂ E generated by x pointwise. Define F (B) = {y ∈ E(B) | there is g : A0 → B such that E(g)(x) = y}. It follows that, for any h: B0 → B, E(h)[F (B0)] ⊂ F (B), and we may define F (h) by restriction. F is clearly a set-valued functor. Since the objects in AbA are additive functors, it follows that F (B) is a subgroup of E(B). Indeed, if y, y0 ∈ F (B), then y = E(g)(x), y0 = E(g0)(x) for some g, g0 : A0 → B, and now y − y0 = E(g)(x) − E(g0)(x) = (E(g) − E(g0))(x) = E(g − g0)(x), so y − y0 ∈ F (B). Thus, F is a group-valued functor. Since x ∈ F (A0), we know F 6= 0. Since i: M → E is essential, we know that F ∩M 6= 0. In particular, there is an object B such that F (B)∩M(B) 6= 0. Let 0 6= y ∈ F (B) ∩ M(B). By the construction of F , there is a morphism h0 : A0 → B such that y = E(h0)(x). Let

49 f A0 A

h0 θ

B ϕ P be a pushout diagram. The pushout theorem gives that ϕ is a monomorphism. Since M is a mono functor, M(ϕ)(y) 6= 0, and so we have 0 6= E(ϕ)(y) = (E(ϕ) ◦ E(h0))(x) = (E(θ) ◦ E(f))(x) = 0, which is a contradiction.

Corollary 6.3.1. Let A be a small abelian category. Then a functor A → Ab may be embedded in an exact functor if and only if it is a mono functor.

Recall that an abelian subcategory A0 of an abelian category A is exact if the inclusion functor A0 → A is exact.

Theorem 6.4 (The ﬁrst embedding theorem). Every small abelian category is isomorphic to an exact full subcategory of Ab. Equivalently, for every small abelian category A, there is an exact embedding functor A → Ab, that is, every abelian category is very abelian.

Proof. Let A be a small abelian category. Consider the group-valued functor G = P Hom(A, ·). Then G is a mono functor. Let E be its injective A∈obj A envelope. By the previous lemma, E is an exact functor, and since G is an embedding functor, so is its every extension. Hence E : A → Ab is an exact embedding functor.

Deﬁnition 6.5. Let M(A) be the full subcategory of AbA whose objects are all the mono functors, or the mono objects. A subobject M 0 of a mono object M is pure in M if the exact sequence

0 M 0 M M/M 0 0 lies in M(A). A mono object is absolutely pure if, whenever it appears as a subobject of a mono object, it is always a pure subobject.

Given a Grothendieck category B, and a full subcategory M closed with respect to subobjects, products and essential extensions, we deﬁne L to be the full subcategory of absolutely pure objects.

Lemma 6.6. If

0 M1 B M2 0

50 is exact and M1,M2 ∈ obj M, then B ∈ obj M.

Proof. Let i: M1 → E be an injective envelope and j : B → E an extension of i. Then B → E ⊕ M2 is a monomorphism. Lemma 6.7. A pure subobject of an absolutely pure object is absolutely pure.

Proof. Let A be absolutely pure, and let P ⊂ A be pure in A. For any monomorphism m: P → M, M ∈ obj M, let R be such that the square P i A m r

M s R is a pushout diagram. Since i and m are monomorphisms, so are s and r. Consider the diagram 0 0 0

p 0 P i A A/P 0

m r f

0 M s R g R/M 0 q h 0 M/P R/A 0 k

0 0, where the two top rows and the two left columns are exact. Our goal is to show that M/P ∼= R/A, as then M will be in M by Lemma 6.6. Since p is a cokernel morphism of i, and g ◦ r ◦ i = g ◦ s ◦ m = 0, there exists a morphism f : A/P → R/M such that f ◦ p = g ◦ r, and similar arguments prove the existence of a morphism k : M/P → R/A such that k ◦ q = h ◦ s. We show that f is an isomorphism, as the claim will then follow from the nine lemma. First, let α: R/M → X be arbitrary. Consider the diagram

51 p P i A A/P

m r f g M s R R/M ϕ α 0 X. Since (α ◦ f ◦ p) ◦ i = (α ◦ f) ◦ (p ◦ i) = 0, we can use the pushout property to get a unique ϕ: R → X such that ϕ ◦ s = 0, ϕ ◦ r = α ◦ f ◦ p = α ◦ g ◦ r, but then the uniqueness of ϕ means that ϕ = α ◦ g. If then β : R/M → X is a morphism such that β ◦ f = α ◦ f, then ϕ = β ◦ g. Since the middle row is exact, the morphism g is an epimorphism, and so β = α, and so f is an epimorphism. To show that f is an isomorphism, it will now suﬃce to show it is a monomorphism. We start by noting that i, m are monomorphisms in a pushout square, so r, s are monomorphisms as well. This makes the square a pullback. Now, consider the diagram P i A f◦p m r

0 M s R g R/M. Since it commutes, the bottom row is exact and the square is a pullback, we obtain that the sequence

f◦p 0 P i A R/M is exact. Now, we apply this to the diagram 0

p 0 P i A A/P 0

1P 1A f f◦p 0 P i A R/M to obtain the exactness of the column, or that f is a monomorphism.

Deﬁnition 6.8. Given an object M in M, we say r : M → R, R ∈ obj L, is a reﬂection of M if, for every morphism f : M → L, there is a unique g : R → L such that the diagram

52 M r R

f g L commutes.

For any object X, let M(X) denote a maximal quotient object of X.

Deﬁnition 6.9. An object T ∈ obj B is a torsion object if, for every M ∈ obj M, Hom(T,M) = 0 or, equivalently, M(T ) = 0.

Theorem 6.10 (The recognition theorem). If the sequence

0 M r R t T 0 is exact in a Grothendieck category B, M mono, R absolutely pure and T torsion, then r is a reﬂection of M in L.

Proof. Consider any f : M → L, L ∈ obj L. Let i: L → E be a monomorphism and E an injective envelope of L, and p: E → F a cokernel of i. Consider the diagram 0 M r R t T 0

f g h 0 L E F 0 i p with exact rows, where g is insured by the injectivity of E, and h is given by the fact that p: E → F is a cokernel of i and t ◦ r = 0, (p ◦ g) ◦ r = (p ◦ i) ◦ f = p ◦ (i ◦ f) = 0. It follows from the choice of g and h that the diagram commutes. By the essential theorem, E is mono, and F is mono because L is absolutely pure. Hence h = 0 and im g ⊂ L, so one obtains a morphism g0 : R → L such that the diagram M r R

f g0 L commutes. The uniqueness of g0 is seen by considering extensions α, β : R → L of F . Denote δ = α − β. Now, δ ◦ g = (α − β) ◦ g = α ◦ g − β ◦ g = f − f = 0, so δ factors through t, but T is torsion and L mono, so δ = 0 and α = β.

53 Theorem 6.11 (The construction theorem). For every mono object M ∈ obj M, there is a monomorphism M → R which is a reﬂection of M in L.

Proof. Embed M in an injective envelope E, and construct the exact commutative diagram 0 0 0

0 M R T 0

0 M E F 0

0 M(F ) M(F ) 0

0 0 by starting with the middle row, then the right column, then the bottom row, and then the top row using the nine lemma. Now, T is torsion and R is a pure subobject of an absolutely pure subobject, and thus absolutely pure itself. It follows that the top row is the exact sequence of the previous theorem.

Theorem 6.12. A mono functor M ∈ obj AbA is absolutely pure if and only if it is left-exact.

Proof. Since M can be embedded in a functor that is both absolutely pure and left-exact, namely its injective envelope, it suﬃces to prove that a pure subfunctor of a left-exact functor is left-exact. Let 0 M E F 0 be exact in AbA, E left-exact and F mono. Let 0 A0 A A00 0 be exact in A. Consider the commutative diagram

54 0 0 0

0 M(A0) M(A) M(A00)

0 E(A0) E(A) E(A00)

0 F (A0) F (A)

0 0. The hypothesis of 2.14 is satisﬁed, so F is mono if and only if M is left-exact.

Corollary 6.12.1. The category Lex = Lex(A, Ab) = L(AbA) is the category of left-exact functors.

We will deﬁne products of functors "pointwise". For any set {Fj | j ∈ J} 0 Q Q Q of functors A → A , deﬁne ( Fj)(A) = Fj(A) and ( Fj)(f) = Q j∈J j∈J j∈J j∈J Fj(f), for any object A and any morphism f in A. Theorem 6.13. Lex is abelian and every object has an injective envelope.

Proof. (0) The 0-functor is in Lex. (1, 1*) For M ∈ obj M = obj M(A, Ab), it is the case that M ∈ obj Lex if and only if the reﬂection r : R → R(M) is an isomorphism. Since the functor R is additive, Lex is closed under the formation of products and sums. A (2) By Lemma 5.7, the Ab -kernel of a morphism f : L1 → L2 in Lex is in Lex, and hence Lex has kernels. Moreover, a morphism in Lex is an AbA-monomorphism if and only if it is a Lex-monomorphism. (3) Given a monomorphism f : L1 → L2 in Lex, let

f g 0 L1 L2 M 0

A be exact in Ab . The absolute purity of L1 assures that M ∈ obj M(A, Ab), and so f is the kernel of r ◦ g, where r : M → R(M) is a reﬂection of M in Lex. (2*) Let f : L1 → L2 be a morphism in Lex, and let

f p L1 L2 F 0

55 A p m r be exact in Ab . Then L2 −→ F −→ M(F ) −→ R(M(F )) is the cokernel of f. (3*) The construction above shows that a morphism f : L1 → L2 in Lex is a Lex-epimorphism if and only if the AbA-cokernel of f is torsion. Let f A be a Lex-epimorphism, and let M be the Ab -image of f and i: M → L2 be a monomorphism, so that

i t 0 M L2 T 0

A is exact in Ab . Since T is torsion, the recognition theorem gives L2 = 0 0 R(M), and so, if k : L0 → L1 is the kernel of f : L1 → M, f = i ◦ f , then the cokernel of k is f, and every Lex-epimorphism is a Lex-cokernel. Since monomorphisms are the same in AbA and in Lex, if E is an AbA- injective envelope of a Lex-object, it is injective in Lex.

Theorem 6.14. Lex is complete and has an injective cogenerator.

Proof. Given a family {Fi}i∈I of left-exact functors, their sum deﬁned in AbA is left-exact, and hence the same as their sum deﬁned in Lex. Since the construction of product is straightforward, Lex is complete. Finally, consider the functor G = Q Hom(A, ·). Now, G is a generator for Lex, A∈obj A and Lemma 3.17 gives the existence of an injective cogenerator.

Theorem 6.15. The functor H : A → Lex, A 7→ Hom(A, ·) = HA, with A a small abelian category, is an exact full embedding.

Proof. By the Yoneda lemma, H is a full embedding. Let 0 A0 A A00 0 be exact in A. The goal is to show that

0 HA00 HA HA0 0 is exact in Lex. Such is the case if and only if the sequence

0 Hom(HA0 ,E) Hom(HA,E) Hom(HA00 ,E) 0 is exact for an injective cogenerator E in Lex. Using the Yoneda lemma, one obtains a commutative diagram

0 Hom(HA0 ,E) Hom(HA,E) Hom(HA00 ,E) 0

0 E(A0) E(A) E(A00) 0,

56 where the vertical arrows are isomorphisms. The sequence is, thus, exact if and only if E is right-exact, but this follows from Lemma 5.14. Recall that an abelian category A is fully abelian if, for every full, small and exact subcategory A0, there is a ring R such that A0 can be embedded fully and exactly into R-mod. For the proof of the next theorem, we consider completeness of a category, as defined in Definition 4.34. One can easily see that left-completeness is dual to right-completeness, and so completeness is a self-dual property. Another remark is that, given any embedding functor F : C → D , one can define D0 to be the subcategory of D the objects and morphisms of which are of the form F (A) and F (f), respectively, with A being an object and f a morphism in C. One can then construct functors F 0 : C → D0 and G: D0 → C by F 0(A) = F (A) and F 0(f) = F (f) for all objects A and morphisms f in C, and G(B) = A and G(g) = f for all objects B and morphisms g in D0, where F (A) = B and F (f) = g. This shows that there is an isomorphism C ∼= D0 of categories. Finally, one can see that a sequence f g 0 A B C is exact in an abelian category A if and only if the sequence gop f op C B A 0 is exact in Aop. Thus, exactness is a self-dual property. Theorem 6.16 (Mitchell’s embedding theorem). Every abelian category is fully abelian. Proof. By the previous theorem, for every small abelian category, there is a full and exact contravariant embedding into a complete abelian category with an injective cogenerator. Taking the dual of the range category, we obtain, by the duality of the axioms of abelian categories, an embedding into an abelian category. By Lemma 4.24, this category contains a projective generator. Moreover, exactness is self-dual, so the dual of the range category is exact. Finally, as completeness is self-dual, we conclude that we have an ambedding into a complete abelian category with a projective generator. Since embeddings induce isomorphisms, the image category is abelian. Since the embedding is full, the exact sequences of objects of the image category in the range category are precisely the exact sequences in the image category, and so the image category is a small, full and exact abelian subcategory. By Theorem 3.31, we can now embed the image category into a category of modules fully and exactly. Since the composite of two such embeddings is again a full and exact embedding, we have proved the theorem.

57 7 Some corollaries of the embedding theorems

Lemma 7.1 (The ﬁve lemma). Consider a commutative diagram with exact rows

∂1 ∂2 ∂3 ∂4 A1 A2 A3 A4 A5

f1 f2 f3 f4 f5

B1 B2 B3 B4 B5 δ1 δ2 δ3 δ4 of R-modules. (1) If f2, f4 are surjective and f5 is injective, then f3 is surjective. (2) If f2, f4 are injective and f1 is surjective, then f3 is injective. (3) If f1 is surjective, f5 injective, and f2, f4 isomorphisms, then f3 is an isomorphism.

Proof. (1) Let b3 ∈ B3 be arbitrary. The goal is to show there is an element a ∈ A3 such that f3(a) = b3. By exactness, (δ4◦δ3)(b3) = 0, so δ3(b3) ∈ ker δ4. Since f4 is surjective, there is an a4 ∈ A4 such that f4(a4) = δ3(b3). Now, by commutativity, (f5 ◦ ∂4)(a4) = (δ4 ◦ f4)(a4) = 0, so ∂4(a4) ∈ ker f5, but f5 is injective, so ∂4(a4) = 0, and we, by exactness, have a4 ∈ ker ∂4 = im ∂3. There is, then, an element a3 ∈ A3 such that ∂3(a3) = a4, and δ3(b3−f3(a3)) = δ3(b3)−(δ3◦f3)(a3) = δ3(b3)−(f4◦∂3(a3)) = δ3(b3)−f4(a4) = δ3(b3) − δ3(b3) = 0, so b3 − f3(a3) ∈ ker δ3 = im δ2, and there is an element b2 ∈ B2 such that δ2(b2) = b3 − f3(a3). Since f2 is surjective, there is an a2 ∈ A2 such that f2(a2) = b2. Now, f3(∂2(a2)+a3) = (f3 ◦∂2)(a2)+f3(a3) = (δ2 ◦f2)(a2)+f3(a3) = δ2(b2)+f3(a3) = b3 −f3(a3)+f3(a3) = b3, so, choosing a = ∂2(a2) + a3, we have f3(a) = b3. (2) Let a3 ∈ ker f3 be arbitrary. Now, (f4 ◦ ∂3)(a3) = (δ3 ◦ f3)(a3) = 0, so ∂3(a3) ∈ ker f4, but f4 is injective, so a3 ∈ ker ∂3 = im ∂2, and there is an element a2 ∈ A2 such that ∂2(a2) = a3. Since (δ2 ◦f2)(a2) = (f3 ◦∂2)(a2) = 0, so f2(a2) ∈ ker δ2 = im δ1, and there is, by the surjectivity of f1, an element a1 ∈ A1 such that (δ1 ◦ f1)(a1) = f2(a2). By commutativity, (f2 ◦ ∂1)(a1) = f2(a2), and, by injectivity, a2 = ∂1(a1), so a3 = ∂2(a2) = (∂2 ◦ ∂1)(a1) = 0, so ker f3 = 0 and f3 is injective. (3) Immediate from (1) and (2). Lemma 7.2 (The snake lemma). Consider the commutative diagram with exact rows α β A1 A2 A3 0

f g h 0 B B B 1 γ 2 δ 3

58 of R-modules. Then there is an exact sequence

ker f ker g ker h ∂ coker f coker g coker h,

−1 −1 where ∂(c) = (p ◦ γ ◦ g ◦ β )(c) for all c ∈ ker h, where p: B1 → coker f is the canonical projection.

Proof. We will prove that ∂ is well deﬁned. Since γ is injective, it is suﬃcient to show that ∂ is independent of the choice of lifting of a3, and that (g ◦ −1 0 β )(a3) ∈ im γ, for all a3 ∈ ker h. Let a2, a2 ∈ A2 be such that β(a2) = 0 0 a3 = β(a2). Then a2 − a2 ∈ ker β = im α, so we get an element a1 ∈ A1 0 −1 0 such that α(a1) = a2 − a2, and so (γ ◦ g)(a2 − a2) = f(a1) ∈ im f, but −1 0 this means that f(a1) ∈ ker p, so 0 = (p ◦ f)(a1) = (p ◦ γ ◦ g)(a2 − a2) = −1 −1 0 (p ◦ γ ◦ g)(a2) − (p ◦ γ ◦ g)(a2), proving the independence. The second −1 −1 claim follows from exactness: (δ ◦g ◦β )(a3) = (h◦β ◦β )(a3) = h(a3) = 0, −1 so (g◦β )(a3) ∈ ker δ = im γ. The rest of this proof is a diagram chase. Theorem 7.3. All diagrammatic lemmas true in all categories of modules are true in every abelian category.

The proof is essentially the same for all diagrammatic lemmas. Therefore, we will show how to generalize the claim (1) of the ﬁve lemma as an example. First, the assumption is equivalent to the claim that the sequences

ij fj 0 ker fj Aj Bj 0, for j = 2, 4, and

f5 p5 0 A5 B5 coker f5 0 are exact, where ij, j = 2, 4 are the canonical inclusion maps and p5 is the canonical projection. The claim is equivalent to the claim that the sequence

i3 f3 0 ker f3 A3 B3 0, is exact. Since this is true in all module categories, and can be experessed using exactness, the claim is true in all fully abelian categories. By Mitchell’s embedding theorem, the claim holds in all abelian categories. In order to generalize Mitchell’s embedding theorem, we introduce the notion of skeletons.

59 Deﬁnition 7.4. Let C be a non-empty category. A skeleton of C is a full subcategory S(C) such that: (1) For every X ∈ obj C, there is an object Y ∈ obj S(C) such that X ∼= Y , (2) If X,Y ∈ obj S(C) and X ∼= Y , then X = Y . The skeleton of an empty category is the category itself.

Consider the full subcategory Setsf of Sets whose objects are all ﬁnite sets, and let n¯ := {k ∈ N | k < n} for each n ∈ N. Let then N be the full subcategory of Setsf such that obj N = {n¯ | n ∈ N}. We claim that N is a skeleton of Setsf . If S is any ﬁnite set, then its cardinality is some n ∈ N, but this is also the cardinality of n¯, so S ∼= n¯, and all the distinct objects of N have distinct cardinalities, so m¯ ∼= n¯ implies m = n, and so m¯ =n ¯.

Lemma 7.5. Every category is equivalent to its skeletons.

Proof. Let C be a category. Since the claim is trivially true for empty categories, we will assume the category is non-empty. Let S(C) be a skeleton of C, and let I : S(C) → C be the inclusion functor. Deﬁne a functor F : C → S(C) by: (1) For all objects A in C, let F (A) ∈ obj S(C) be the object with A ∼= F (A), and let ϕA : F (A) → A be a ﬁxed isomorphism. (2) For all morphisms f : A → B in C, we let F (f): F (A) → F (B) be the morphism induced by the isomorphisms. Now, F is indeed a full functor, and we claim that I ◦ F ' 1C,F ◦ I ' 1S(C). The latter equivalence holds trivially, since F ◦I = 1S(C). For every object A in C, let ϕA : F (A) → A be the isomorphism used in the construction of F . We claim that, for every morphism f : A → B in C, the diagram

(I◦F )(f) (I ◦ F )(A) (I ◦ F )(B)

ϕA ϕB A B f commutes. First, noticing that (I ◦ F )(A) = F (A) and (I ◦ F )(f) = F (f) −1 for all objects and morphisms in C, we have (I ◦ F )(f) = ϕB ◦ f ◦ ϕA, so −1 ϕB ◦ (I ◦ F )(f) = ϕB ◦ ϕB ◦ f ◦ ϕA = f ◦ ϕA, proving the commutativity. Therefore, we can set ϕ = {ϕA | A ∈ obj C} to get a natural transformation I ◦ F → 1C. Since all the morphisms in ϕ are isomorphisms we have I ◦ F ' 1C, and so C ' S(C). Lemma 7.6. The skeletons of an abelian category A are abelian.

60 Proof. Clearly, any object isomorphic to a zero object (or a product or sum, respectively) is a zero object (or a product or sum, resp.) itself. Therefore every S(A) satisfies the axioms (A0), (A1) and (A1*). Using the functor F defined in the previous lemma, we show that F preserves kernels and cokernels. This is true because the use of isomorphisms in the definition of F makes F preserve uniqueness, and so ker F (f) ∼= F (ker f), coker F (f) ∼= F (coker f) for all morphisms f in A. Finally, we show that the skeletons satisfy the axiom (A3), as the axiom (A3*) is proved similarly. Let f : A → B be a monomorphism in S(A). Then there are objects A0,B0 and a morphism f 0 : A0 → B0 in A such that f = F (f 0). Since 0 ∼= ker f ∼= ker F (f 0) ∼= F (ker f 0) ∼= ker f 0, we see that f 0 is a monomorphism. Therefore, there is some morphism g0 : B0 → C0 such that f 0 is a kernel of g0, and so f is a kernel of F (g0).

Deﬁnition 7.7. A category is called essentially small if its skeletons are small categories.

Clearly, every small category is essentially small. On the other hand, the converse does not hold. Let Sing be the full subcategory of Sets consisting of all singletons. Then Sing is not a small category itself, as {S} is a set for every set S, and otherwise the class of all sets would be a set itself, but a skeleton of Sing consist of just one object, so it is essentially small.

Lemma 7.8. Let A be an essentially small abelian category. Then A is equivalent to a small abelian category B.

Theorem 7.9. For an essentially small abelian category A, there is a unitary ring R such that A is equivalent to a full and exact subcategory of R-mod.

Proof. Suppose B is a small category equivalent to A. By Lemma 7.8, B can be assumed to be abelian, and so there is a unitary ring R such that there is an embedding M : B → R-mod. Let MB be the image category of M, and let M 0 : B → MB be the functor such that the diagram B M R-mod

M 0 I MB commutes, where I : MB → R-mod is the inclusion functor. We claim that M 0 is an equivalence, as this will prove the claim. Deﬁne a functor N : MB → B for objects in MB by N(X) = B, where B ∈ obj B is such that M(B) = X, and, for morphisms in MB, N(f) = g where M(g) = f. Since M is full and surjective, such objects and morphisms always

61 exist, and they are unique because M is an embedding. To see that N is a functor, consider any composite g ◦ f in MB, and now N(g) ◦ N(f) = g0 ◦ f 0, for some morphisms f 0, g0 in B, so M(g0 ◦ f 0) = M(g0) ◦ M(f 0) = g ◦ f. But, by uniqueness, N(g ◦ f) = N(g) ◦ N(f), and, similarly, N(1X ) = 1B. Now, clearly N ◦ M = 1B and M ◦ N = 1MB, so B ' MB. By transitivity, we have A ' MB, as desired. Finally, we present a way to classify essentially small abelian categories using topological spaces. We omit the proof of the existence of the so called Eilenberg-MacLane spaces, that is, spaces K(G, n) where G is any group and n is a positive integer, such that ( ∼ G, if k = n πk(K(G, n)) = 0, otherwise.

For simplicity, we will ﬁx n = 2. Another proof we will omit is that of the existence of a Grothendieck group K0(S) for any commutative semigroup S. With these facts, we will prove the following theorem. A construction of K(A, n) for n ≥ 2 and A (necessarily) abelian can be found in [Cohen, p. 32-33]. The existence of a Grothendieck group of a commutative semigroup S is given by [Rosenberg, 1.1.3].

Theorem 7.10. For every essentially small abelian category, one can assign a topological space unique up to weak homotopy equivalence.

Proof. Let A be an essentially small abelian category, and let R be a unitary ring such that there is, for some skeleton, a Mitchell embedding m: S(A) → R-mod. Since this induces an isomorphism S(A) ∼= mS(A) = R, we conclude that R is abelian. Moreover, R is small. Defining a relation ∼ in R by M ∼ N if and only if M ∼= N, we get an equivalence relation and thus a small category S = R/ ∼. We set S = obj S and claim that S equipped with the direct sum of modules is a commutative monoid. This can be seen easily from the facts that direct sums are associative and commutative, and the trivial module 0 is an identity element, up to isomorphism, and so we obtain the equations [M1] ⊕ ([M2] ⊕ [M3]) = ([M1] ⊕ [M2]) ⊕ [M3], [M1] ⊕ [M2] = [M2] ⊕ [M1] and 0 ⊕ [M1] = [M1] = [M1] ⊕ 0, for all left R-modules M1,M2,M3. Let Sf denote the set of all the equivalence classes of finitely generated left R-modules, we note that M ⊕ N is finitely generated for any finitely generated M,N, so it follows that Sf is closed under the operation. Hence, Sf is a submonoid of S.

62 As a submonoid of a commutative monoid, Sf is, itself, commutative, and monoids are semigroups, so we can deﬁne G = K0(Sf ). Setting XA = K(G, 2), we obtain a topological space unique up to weak homotopy equivalence. Finally, we will show that this space is independent of the choice of skeleton up to weak homotopy equivalence. If S0(A) is another skeleton, then there is an isomorphism S(A) ∼= S0(A), and this induces an isomor- ∼ 0 phism Sf = Sf of the subcategories of the equivalence classes of ﬁnitely ∼ 0 0 0 ∼ generated modules. It follows that Sf = Sf , and so G = K0(Sf ) = G, from 0 which we obtain that XA is weakly homotopy equivalent to K(G , 2).

8 References and bibliography

[Freyd] Freyd, P.: Abelian Categories, Joanna Cotler Books, 1966. [Awodey] Awodey, S.: Category Theory (2nd edition), Oxford Logic Guides, 2010. [Cohen] Cohen, J. M.: Stable Homotopy, Springer-Verlag, 1970. [Rosenberg] Rosenberg, J.: Algebraic K- theory and Its Applications, Springer, 1994.