Simplicial torsors (Q2759194)

\textit{J. Duskin} [in: Applications of sheaves, Lect. Notes Math. 753, 255-279 (1979; Zbl 0444.18014)] defined a notion of simplicial torsor, which provided a combinatorial interpretation of \(H^n(\mathcal{E}, A)\), the \(n\)th cohomology group of a Grothendieck topos \(\mathcal{E}\) with coefficients in an abelian group object, \(A\) in \(\mathcal{E}\). The development of his theory was accomplished by \textit{P. G. Glenn} [J. Pure Appl. Algebra 25, 33-105 (1982; Zbl 0487.18015)]. Their interpretation of this abelian sheaf cohomology was as the connected components of a category of `simplicial torsors' over the Eilenberg-MacLane object \(K(A,n)\). NEWLINENEWLINENEWLINEIn this beautiful paper, the author examines what happens if the base for the torsors is not necessarily a \(K(A,n)\). The development of the theory of simplicial sheaves since the 1980s makes his task one of collecting the pieces of the jigsaw, turning them picture side up and then putting them together. The result provides an excellent overview of a variety of interlocking pieces of theory, comparing the advantages of different approaches as it progresses.