Higher cohomologies of modules

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 ${Ext}_{{Mod}_{ℂ}}^{n} (B, A)$ provide the basic cohomology theory for $ℂ$ –modules. We introduce, for each integer $r \geq 1$ , an approach for a level– $r$ cohomology theory for $ℂ$ –modules by defining cohomology groups $H_{[b] ℂ, r}^{n} (B, A)$ , $n \geq 0$ , 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.

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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

References

Publication

Received: 3 February 2011

Revised: 7 November 2011

Accepted: 18 November 2011

Published: 14 March 2012

Authors

María Calvo
Department of Algebra Faculty of Sciences University of Granada Campus Fuentenueva 18071 Granada Spain

Antonio M Cegarra
Department of Algebra University of Granada Campus Fuentenueva, Faculty of Sciences 18071 Granada Spain

Nguyen T Quang
Department of Mathematics Hanoi National University of Education Xuan Thuy Street, Cau Giay district Hanoi 136 Vietnam

Volume 12, issue 1 (2012)

María Calvo, Antonio M Cegarra and Nguyen T Quang

Abstract