Crossed Simplicial Group Categorical Nerves

TL;DR

This paper extends the nerve construction for categories within crossed simplicial groups, introduces twisted versions, and applies them to develop equivariant derived stacks.

Contribution

It provides explicit descriptions of nerves for crossed simplicial groups and introduces twisted nerves for new equivariant geometric applications.

Findings

Explicit generators and relations for nerves of crossed simplicial groups

Development of twisted nerve constructions for equivariant applications

Application to equivariant derived stacks

Abstract

We extend the notion of the nerve of a category for a small class of crossed simplicial groups, explicitly describing them using generators and relations. We do this by first considering a generalised bar construction of a group before looking at twisted versions of some of these nerves. As an application we show how we can use the twisted nerves to give equivariant versions of certain derived stacks.