Crossed simplicial group categorical nerves (Q722249)

In this article, the author addresses the question of finding ``meaningful'' extensions of the simplex category \(\Delta\), whose objects are the non-empty finite ordered sets \([n]=\{1,\dots,n\}\) and morphisms the order preserving maps. Specifically, the author seeks an extension with properties similar to Connes' cyclic category [\textit{A. Connes}, Noncommutative geometry. Academic Press (1994; Zbl 0818.46076)]. Hence, the author gives an explicit construction of a crossed simplicial version of the nerve and of the bar construction of a group for some particular examples of crossed simplicial groups \(\Delta\mathfrak G\), resulting from a classification theorem. Namely there are seven of them, which the author calls \textit{simple crossed simplicial groups}. In the last section of the article, the author investigates the equivariant derived stack of local systems.