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 2 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 2 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 2 P the induced map C(P; M) ! C(P; N) is surjective. (3) For every M 2 obC there is morphism P ! M in E such that P 2 P. Definition. A morphism f : M ! N in a category C is a (categorical) epimorphism if for every W 2 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 2 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..