Category cohomology

Let be a small category, an Abelian category with exact infinite products, and a covariant functor. Define the objects for in the following way:

where is a sequence of morphisms of with , . Let

be the monomorphism induced by the family of morphisms

and by the family

Here, denotes the projection ).

The morphisms , satisfy the conditions , and therefore one obtains a complex in . The homology objects of this complex are called the cohomology of with coefficients in and are denoted by . For any functor there are a functor (called the co-induced functor) and a monomorphism such that for .

The functor is an exact functor. Therefore, any short exact sequence of functors induces a long exact sequence of cohomology objects of . It can be proved that the functors form a universal connected (exact) sequence of functors and that

where is th right satellite of the functor (here, denotes the category of functors from to , and is an (inverse) limit functor).

For a small category , let denote the pre-additive category whose objects are those of and is the free Abelian group on (cf. also Free group). Composition is defined in the unique way so as to be bilinear and to make the inclusion a functor. If is a monoid, then is the monoid ring of with coefficients in . The inclusion induces an isomorphism of categories

where the left-side is the category of additive functors from to (the category of Abelian groups) and the right-hand side side is the category of all functors from to .

If is the category of Abelian groups (), one has

where denotes the constant functor (i.e. , for any object and any morphism of ), and is taken in the category of additive functors .

If is a group (i.e. a category with one object and whose morphisms are invertible) and , then the groups are the cohomology groups (cf. also Cohomology group) of the group with coefficients in , which is a module over the group ring (cf. also Cross product). In this case the co-induced functor is a co-induced -module .

As for the case of groups, -fold extensions of categories can be defined and the isomorphism with cohomologies of categories can be established. Under additional assumptions, the properties of group cohomologies are obtained for category cohomologies (e.g. the universal coefficient formula for the cohomology group of a category, etc.).

For a commutative ring , a -category is a category equipped with a -module structure on each hom-set in such a way that composition induces a -module homomorphism. If , a -category is just a pre-additive category. B. Mitchell has defined the (Hochshild) cohomology group of a small -category with coefficients in a bimodule (i.e., bifunctor) (where denotes the tensor product of categories). H.-J. Baues and G. Wirshing have introduced cohomology of a small category with coefficients in a natural system, which generalizes known concepts and uses Abelian-group-valued functors (i.e. modules) and bifunctors as coefficients.

References

[a1]	H.-J. Baues, G. Wirshing, "Cohomology of small categories" J. Pure Appl. Algebra , 38 (1985) pp. 187–211
[a2]	T. Datuashvili, "The cohomology of categories" Tr. Tbiliss. Mat. Inst. A. Razmadze, Akad. Nauk Gruzin.SSR , 62 (1979) pp. 28–37
[a3]	G. Hoff, "On the cohomology of categories" Rend. Math. (VI) , 7 : 2 (1974) pp. 169–192
[a4]	M.J. Lee, "A generalized Mayer–Vietoris sequence" Math. Jap. , 19 : 1 (1974) pp. 41–50
[a5]	B. Mitchell, "Rings with several objects" Adv. Math. , 8 (1972) pp. 1–161
[a6]	D.G. Quillen, "Higher algebraic -theories" H. Bass (ed.) , Algebraic -theory I , Lecture Notes Math. , 341 , Springer (1973) pp. 85–147
[a7]	J.-E. Roos, "Sur les foncteurs derives de . Applications" C.R. Acad. Sci. Paris , 252 (1961) pp. 3702–3704
[a8]	Ch.E. Watts, "A homology theory for small categories" , Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) , Springer (1966) pp. 331–335