Higher cohomologies of modules

Abstract

If $ℂ$ is a small category, then a right $ℂ$–module is a contravariant functor from $ℂ$ into abelian groups. The abelian category ${Mod}_{ℂ}$ of right $ℂ$–modules has enough projective and injective objects, and the groups ${Ext}_{{Mod}_{ℂ}}^n\left(B,A\right)$ provide the basic cohomology theory for $ℂ$–modules. We introduce, for each integer $r\ge 1$, an approach for a level–$r$ cohomology theory for $ℂ$–modules by defining cohomology groups $H_{\left[b\right]ℂ,r}^n\left(B,A\right)$, $n\ge 0$, which are the focus of this article. Applications to the homotopy classification of braided and symmetric $ℂ$–fibred categorical groups and their homomorphisms are given.