Second cohomotopy and nonabelian cohomology

Abstract

The main difficulty in the theory of non-abelian cohomology is that for cosimplicial groups only zero-th and first dimensional cohomotopy are known. In this article we introduce a new class of cosimplicial groups, called centralised cosimplicial groups, for which we are able to define a second cohomotopy, with all expected properties. The main examples of such cosimplicial groups come from 2-categories.

Schreier Theory of Track Categories

“…This paper is a continuation of our study of non-abelian Baues-Wirsching cohomologies. In our previous paper [3], we defined second non-abelian cohomology H 2 (C, D) of a small category C with coefficients in a so-called centralised natural system D. We proved that H 2 (C, D) classifies linear extensions of C by D, generalising the corresponding result for abelian natural systems proved in [2].…”

The motivic anabelian geometry of local heights on abelian varieties

“…This generalisation is provided by the theory of cosimplicial groups and their cohomotopy, as defined for instance in [BK72], and which subsumes the more well-known theory of non-abelian group cohomology [Ser97, Section I.5]. For the convenience of the reader, we will recall the basic setup here as expounded in [Pir14].…”