Cohomology of cofibred categorical groups
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. Hardie}, \textit{K. H. Kamps} and \textit{R. W. Kieboom}, ``A homotopy bigroupoid of a topological space'', Appl. Categ. Struct. (to appear)]. The principle algebraic idea is to extend the Schreier analysis of extensions of groups to a cohomological classification of \({\mathcal B}\)-torsors under a \({\mathcal B}\)-categorical group \(\mathbb{G} \). (A \({\mathcal B}\)-torsor under \(\mathbb{G}\) is a cofibration \({\mathcal P}_{\mathcal E}: {\mathcal E} \to {\mathcal B}\) endowed with a \({\mathcal B}\)-action of \(\mathbb{G}\) on \({\mathcal E}\) such that for any object \(A\) of \({\mathcal B}\) the fibre category \({\mathcal E}_A\) is equivalent to the categorical group \(\mathbb{G}_A\) via the given action.) This leads to the cohomology \(\mathbb{H}^i ({\mathcal B},\mathbb{G})\), \(0\leq i\leq 2\) (a pointed set for \(i=2\), a group for \(i=1\), an abelian group for \(i=0)\). The paper contains an explicit interpretation of the cohomology in a wide range of algebraic and topological examples including Baues' theory of linear extensions of a small category [\textit{H. J. Baues}, ``Algebraic homotopy'', Camb. Stud. Adv. Math. 15 (1989; Zbl 0688.55001); \textit{H.-J. Baues} and \textit{G. Wirsching}, ``The cohomology of small categories'', J. Pure Appl. Algebra 38, 187-211 (1985; Zbl 0587.18006)].
scientific article; zbMATH DE number 1384146