#### Volume 12, issue 1 (2012)

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Higher cohomologies of modules

### María Calvo, Antonio M Cegarra and Nguyen T Quang

Algebraic & Geometric Topology 12 (2012) 343–413
##### 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.

##### Keywords
module, simplicial set, Eilenberg–Mac Lane complex, homotopy colimit, cohomology, fibred braided monoidal category
##### Mathematical Subject Classification 2010
Primary: 18D10, 55N25
Secondary: 55P91, 18D30