Cohomology of cofibred categorical groups
type: Item
instance of: scholarly article
title: Cohomology of cofibred categorical groups
zbMATH Open document ID: 0944.18011
DOI: 10.1016/S0022-4049(98)00109-1
author: Yanyan Li
published in: Journal of Pure and Applied Algebra
publication date: 2000-09-18T00:00:00Z
review text: The paper is concerned with a certain kind of non-abelian cohomology \(\mathbb{H}^i({\mathcal B},\mathbb{G})\), \(0\leq i\leq 2\), defined for a small category \({\mathcal B}\) with coefficients in a \({\mathcal B}\)-(cofibred) categorical group \(\mathbb{G}\), i.e., with coefficients taken as bundles of categorical groups, instead of bundles of abelian groups. This means we have a \({\mathcal B}\)-monoidal category \(\mathbb{G}\), i.e., a monoidal category \(\mathbb{G}\) relative to a functor \({\mathcal P}_\mathbb{G}: \mathbb{G}\to {\mathcal B}\) which is required to be a cofibration in the sense of Grothendieck [\textit{A. Grothendieck}, ``Catégories fibrées et descente'', SGA 1, exposé VI, Lect. Notes Math. 224, 145-194 (1971; Zbl 0234.14002)] such that every fibre category \(\mathbb{G}_A\), where \(A\) is an object of \({\mathcal B}\), is a categorical group, i.e., a monoidal groupoid in which each object has a quasi-inverse with respect to the tensor product. \({\mathcal B}\)-categorical groups arise in numerous algebraic and topological problems such as the classification of arbitrary extensions of groups, the classification of Azumaya algebras over a commutative ring, or the classification of homotopy classes of cross-sections for a fibration over an Eilenberg-MacLane space of type \(K(G,1)\). Among the topological examples the authors construct what they call the cofibred Whitehead categorical group generalising the Whitehead crossed module from the second relative homotopy group to the fundamental group [\textit{J. H. C. Whitehead}, ``Combinatorial homotopy. II'', Bull. Am. Math. Soc. 55, 453-496 (1949; Zbl 0040.38801)], the Whitehead 2-groupoid studied by Moerdijk and Svensson [\textit{I. Moerdijk} and \textit{J.-A. Svensson}, ``Algebraic classification of equivariant homotopy 2-types. I,'' J. Pure Appl. Algebra 89, No. 1-2, 187-216 (1993; Zbl 0787.55008)] and the homotopy bigroupoid of a space studied by Hardie, Kieboom and the reviewer [\textit{K. A. Ha…
Mathematics Subject Classification ID: 18G50
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16K50
zbMATH DE Number: 1384146
zbMATH Keywords: Galois extension
zbMATH Keywords: torsors
zbMATH Keywords: non-abelian cohomology
zbMATH Keywords: categorical group
zbMATH Keywords: cofibration
zbMATH Keywords: Eilenberg-MacLane space
zbMATH Keywords: crossed module
zbMATH Keywords: 2-groupoid
zbMATH Keywords: homotopy bigroupoid
zbMATH Keywords: linear extensions
reviewed by: Klaus Heiner Kamps
MaRDI profile type: Publication
full work available at URL: s0022-4049(98)00109-1
OpenAlex ID: W2058687745
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description: scientific article; zbMATH DE number 1384146