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1. Abelian Categories Most of Homological Algebra Can Be Carried

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1. Abelian categories Most of homological algebra can be carried out in the setting of abelian categories, a class of categories which includes on the one hand all categories of modules and on the other hand categories of (abelian) sheaves. In this section we give a quick introduction to abelian categories. There are three layers of structure, and we discuss each in succession. 1.1. Pre-additive categories. In a general category C the homsets C(A; B) are simply sets; however, in many specific cases we find that these sets carry some additional structure. For example, consider the category Pos of posets and order-preserving functions. For two posets A; B, the homset Pos(A; B) is again an ordered set if we let f ≤ g , f(x) ≤ g(x) for all x 2 A. Sim- ilarly, in the category Rel of sets and relations we may define an ordering on Rel(A; B) by saying that R ≤ S if and only if R (as a subset of A × B) is contained in S, i.e. if aRb implies aSb. Also, in the category of abelian groups, the homset AbGrp(A; B) is more than a mere set: it is itself an abelian group under the operation of pointwise addition. More generally, in any category of modules we have that R − Mod(A; B) is an abelian group. The phenomenon that categories may have homsets with additional struc- ture goes by the name of enrichment: we say that the category Rel is en- riched in posets, or that the category of R-modules is enriched in abelian groups. We will not go into the theory of enriched categories in general (a standard reference is Max Kelly's book \Enriched Category Theory") but only concentrate on the abelian group case. Definition 1.1. A category C is called pre-additive (equivalently: enriched in abelian groups) if for each pair of objects A; B of C the homset C(A; B) is an abelian group; moreover, it is required that the composition functions C(B; C) × C(A; B) −! C(A; C) are additive in each variable. Explicitly, the condition on composition means that given a diagram g f / k A / B / C / D h we have (g + h)f = gf + hf; k(g + h) = kg + kh: Note that pre-additivity is really an extra piece of structure on the cat- egory: in particular, a category may be pre-additive in different ways. To see this, take a category with two objects A; B and let Hom(A; A) = 0 = Hom(B; B), and Hom(A; B) a 4-element set. Then since we have ∼ Z=4Z =6 Z=2Z ⊕ Z=2Z there are two different abelian group structures on Hom(A; B). Examples 1.2. (1) As mentioned earlier, any category of R-modules is pre-additive when we define a pointwise group structure on the homsets. In particular the category of abelian groups is pre-additive. 1 2 (2) Recall that a one-object category is the same thing as a monoid. Now a one-object pre-additive category is the same thing as a monoid which also has the structure of an abelian group, in such a way that the multiplication (which is the composition in the one-object category) preserves addition. Such a structure is commonly called a ring. So, if you like, you can regard a pre-additive category as a \multi-object" ring. (3) If C is pre-additive then so is the opposite category Cop. (4) If C is pre-additive then so is any functor category CD. Exercise 1. Work out the last example. Exercise 2. Show that pre-additive structure on a category C corresponds to a lift of the Hom-functor of C, as in AbGrp s9 s s U s s Cop × C / Set C(−;−) where U is the forgetful functor. Definition 1.3 (Additive functor). Let F : C −! D be a functor between pre-additive categories. We say F is additive whenever for each pair of objects A; B of C the induced function C(A; B) −! D(F A; F B) is an additive map. The apparent terminological mismatch (pre-additive vs. additive) will be cleared up in the next section. Examples 1.4. (1) The forgetful functor R − Mod −! AbGrp is additive. (2) When C and D are both pre-additive categories, then we may con- sider the (full) subcategory of CD on those objects which are additive functors C −! D. This is again a pre-additive category. (3) Let R be a ring, considered as one-object pre-additive category. By the previous example the category of additive functors from R to the category of abelian groups is again a pre-additive category. (4) For any pre-additive category with terminal object, the global sec- tions functor factors additively through the category of abelian groups. Exercise 3. Show that global sections is an additive functor. Exercise 4. Prove that the category of additive functors from R to AbGrp is equivalent (via an additive functor!) to the category of R-modules. Exercise 5. Let A be an object of a pre-additive category C. Show that End(A) = C(A; A) is a ring. 3 1.2. Additive categories. We now add some extra structure to our pre- additive categories: Definition 1.5 (Additive category). A pre-additive category C is additive if it has finite products and a zero object. Most examples of pre-additive categories given before are in fact additive: for any ring R, the category of R-modules is additive; if C is additive then so is CD for any D, and so is the subcategory on the additive functors D −! C if D is also (pre-)additive. The correct notion of functor between additive categories is the same as for pre-additive categories. The reason is that, as we shall see below, any additive functor automatically preserves finite products and zero objects. We now show that the notion of additive category is, in spite of the apparent non-duality of the definition, is self-dual. For that, we show that any additive category not only has products but also has finite coproducts. In fact: Lemma 1.6. If C is additive, then C has biproducts. Proof. Given two objects A; B write A ⊕ B for the product of A and B. We shall show that A ⊕ B is (the vertex of) a coproduct of A and B. Define coproduct inclusions by h1A; 0i : A −! A ⊕ B; h0; 1Bi : B −! A ⊕ B: Given morphisms f : A −! C and g : B −! C define [f; g]: A ⊕ B −! C as fπA + gπB. Remaining details are left as exercise. Exercise 6. Complete the proof. We now obtain: Corollary 1.7. If C is additive then so is Cop. Note that we used that additive structure on homsets (and the zero maps) to define the coproduct structure. Thus, the additive structure on C and the finite products are related! To make this clearer, consider two parallel maps f; g : A −! B. On the one hand, since our category is pre-additive, we may use the abelian group structure on C(A; B) to form f + g. But on the other hand we can consider [f; g] A −−!δ A ⊕ A −−−−! B: Here, δ is the diagonal map. From the construction of [f; g] it is clear that f + g = [f; g]δ. Thus, the addition on C(A; B) can be recovered from the biproduct structure. Moreover, the map A −! 0 −! B is indeed the unit element of C(A; B) (check!). This means that in an additive category the pre-additive structure (i.e. the abelian group structures on C(A; B) is uniquely determined. Conse- quently, unlike for pre-additive categories, it is impossible for a category to be additive in more than one way. Exercise 7. Let F : C −! D be an additive functor between additive cate- gories. Show that F preserves finite products and the zero object. 4 1.3. Abelian categories. The final step to make is to add to the notion of additive category some exactness properties: Definition 1.8 (Abelian Category). An additive category C is called abelian if: (i) every map in C has a kernel; (ii) every map in C has a cokernel; (iii) every monomorphism in C is the kernel of its cokernel; (iv) every epimorphism in C is the cokernel of its kernel. Note that this definition is again self-dual so that C is abelian if and only if Cop is. The conditions on monomorphisms and epimorphisms imply that these are regular; in particular, an abelian category is balanced, in the sense that any morphism which is at the same time mono and epi is an isomorphism. Examples 1.9. (1) The leading example of an abelian category is R − Mod for any ring R. (2) If C is abelian then so is CD for any D. (3) If C is abelian and D is (pre-)additive then the category of additive functors D −! C is again abelian. (Thus the fact that R − Mod is abelian is a consequence of the fact that this category is of the form AbGrpR and that AbGrp is abelian.) (4) The category of finite abelian groups is abelian. (5) The category of free abelian groups is additive but not abelian (check!). We begin with a few easy consequences of the definition. Lemma 1.10. In an abelian category, every morphism factors as an epi- morphism followed by a monomorphism.
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