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

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

module, simplicial set, Eilenberg–Mac Lane complex, homotopy colimit, cohomology, fibred braided monoidal category
Mathematical Subject Classification 2010
Primary: 18D10, 55N25
Secondary: 55P91, 18D30
Received: 3 February 2011
Revised: 7 November 2011
Accepted: 18 November 2011
Published: 14 March 2012
María Calvo
Department of Algebra
Faculty of Sciences
University of Granada
Campus Fuentenueva
18071 Granada
Antonio M Cegarra
Department of Algebra
University of Granada
Campus Fuentenueva, Faculty of Sciences
18071 Granada
Nguyen T Quang
Department of Mathematics
Hanoi National University of Education
Xuan Thuy Street, Cau Giay district
Hanoi 136