Higher cohomologies of modules

Calvo, María¹ ; Cegarra, Antonio M² ; Quang, Nguyen T³

¹Department of Algebra, Faculty of Sciences, University of Granada, Campus Fuentenueva, 18071 Granada, Spain
²Department of Algebra, University of Granada, Campus Fuentenueva, Faculty of Sciences, 18071 Granada, Spain
³Department of Mathematics, Hanoi National University of Education, Xuan Thuy Street, Cau Giay district, Hanoi 136, Vietnam

Algebraic and Geometric Topology, Tome 12 (2012) no. 1, pp. 343-413

Résumé

If ℂ is a small category, then a right ℂ–module is a contravariant functor from ℂ into abelian groups. The abelian category Modℂ of right ℂ–modules has enough projective and injective objects, and the groups ExtMod ℂn(B,A) provide the basic cohomology theory for ℂ–modules. We introduce, for each integer r ≥ 1, an approach for a level– r cohomology theory for ℂ–modules by defining cohomology groups H[b]ℂ,rn(B,A), n ≥ 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.

Détail

DOI : 10.2140/agt.2012.12.343

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

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