Preadditive and additive categories
A category is preadditive if every morphism set, (A, B), has an abelian group
structure such that composition on either side is linear. A preadditive category with
one object is simply a ring. An additive category is one which is preadditive,
has a zero (=initial and final) object and which is such that every pair of objects
has a coproduct: in that case every pair of objects has a direct product, which is
canonically isomorphic to their coproduct and which is referred to as their direct
sum (see, e.g., [77, 2.1.2]). An additive category is abelian if every morphism has a
kernel and a cokernel, if every monomorphism is a kernel and if every epimorphism
is a cokernel (one says then that every monomorphism and every epimorphism is
regular). The archetype is the category, Ab, of abelian groups. The category of
finitely generated abelian groups also is an example.
Start with a small preadditive category A. Denote by
the additive com-
pletion of A ([29, p. 60], [76, p. 92]). The objects of
are finite tuples of objects
of A (including the empty tuple, i.e. the zero object of A+) and maps are matrices
with morphisms from A as entries. This has the universal property that for all
additive functors A −→ B with B additive there is a unique, up to natural equiv-
alence, factorisation through the canonical functor A −→ A+ which takes objects
to 1-tuples. (A functor F : A B is additive if for every A, A A the map
F : (A, A ) (FA, FA ), f Ff is a homomorphism of abelian groups.)

Digression: unique versus unique to natural equivalence. Take A = F2: a category
with one object, whose endomorphism ring is the field of two elements. Then
is a skeletal version of the category of finite-dimensional F2-vectorspaces. (A cat-
egory is skeletal if there is just one object in each isomorphism class.) Replacing
F2 by an arbitrary ring A = R, we obtain the category of finitely generated free
right R-modules, at least a skeletal version but we seldom make such distinctions.
Let B be a version of
with two copies of each vector space and define additive
functors F, G :
−→ B to agree on 0 and the 1-dimensional space but to disagree
on higher-dimensional spaces. Clearly there is a natural equivalence (even isomor-
phism) between F and G, induced by the obvious corresponding automorphism of
B, but certainly F = G. Thus the extension of A −→ B to
−→ B, though
unique to natural equivalence, i.e. to isomorphism in the functor category (A+, B),
is not literally unique. (We use the notation (A, B) for the category of additive
functors from the preadditive category A to the preadditive category B. Usually
we assume that A is skeletally small, that is, has, up to isomorphism, only a set
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