Projective classes

Definition. A pre-additive category is a category C together with a choice of an abelian group structure on C(M,N) for each M,N ∈ obC with the property that each composition map is bilinear. For example, the category of abelian groups is pre-additive. So is the category of non-trivial abelian groups. In a pre-additive category, there is a “zero” morphism between any two ojects. Definition. Let C be a pre-additive category and f : M → N. A morphism i : K → M is a kernel for f if (1) f ◦ i = 0, and (2) any map j : L → M such that f ◦ j = 0 factors uniquely through i. A pre-additive category C has kernels if every morphism in C has a kernel. If i : K → M and j : L → M are both kernels for f : M → N, then there is a unique map k : L → K such that j = i ◦ k, a unique map l : K → L such that i = j ◦ l, and k and l are inverse isomorphisms. Thus we speak of the kernel, if one exists. For example, any zero map 0 : X → Y has a kernel, given by the identity map 1X : X → X. Definition. Let C be a pointed category with kernels. A projective class in C is a pair (P, E), where P is a class of objects in C and E is a class of morphisms in C, such that (1) P ∈ obC lies in P if and only if for every f : M → N in E the induced map C(P,M) → C(P,N) is surjective. (2) f : M → N lies in E if and only if for every P ∈ P the induced map C(P,M) → C(P,N) is surjective. (3) For every M ∈ obC there is morphism P → M in E such that P ∈ P. Definition. A morphism f : M → N in a category C is a (categorical) epimorphism if for every W ∈ obC the induced map C(N,W ) → C(M,W ) is injective. An object P in C is a (categorical) projective if for every epi- morphism M → N in C, the induced map C(P,M) → C(P,N) is surjective. The category C has enough (categorical) projectives if for every M ∈ obC there is a categorical epimorphism P → M with P a categorical projective. Lemma. If C is a pre-additive category with kernels and enough projectives, then the categorical projectives and the categorical epimorphisms form a projective class, the categorical or absolute projective class. Definition. Let C be a preadditive category with kernels, and let (P, E) be a projective class in C. A diagram

f g M 0 ←− M ←− M 00 is exact (relative to the projective class) if f ◦ g = 0 and the canonical map M 00 → ker f is in E. Definition. Let C be a pointed category with kernels, and let (P, E) be a projective class in C. A diagram

0 ←− M ←− P0 ←− P1 ←− P2 ←−· · · is a resolution of M (relative to the projective class) if it is exact at each joint and each Pn lies in P. Theorem. Let C be a pre-additive category with kernels, and let (P, E) be a projective class in C. (1) Any object M admits a projective resolution. (2) Given a diagram

0 ←− M ←− P0 ←− P1 ←− P2 ←−· · · in which each composite is 0 and each Pn lies in P, and a diagram

0 0 0 0 0 ←− M ←− P0 ←− P1 ←− P2 ←−· · · which is exact at each joint, and a map f : M → M 0, there is a chain map 0 f∗ : P∗ → P∗ covering f. (3) This chain map is unique up to chain homotopy. Example. Let C be a category, let Ab denote the category of abelian groups, and let Fun(C, Ab) be the category of functors from C into Ab. (We will ignore potential set theoretic objections—the morphisms in this category may not form sets—since they do not interfere with what we are doing.) Any set M of objects—to be called “models”—of C determines a projective class in Fun(C, Ab), with the following definitions: An object is M-projective if it is a retract of a the free abelian group of a coproduct of functors representable by models. A natural transformation θ : F → G is M-epi if θM : F (M) → G(M) is surjective for every model M. Lemma. These definitions constitute a projective class.