Equivalence of models for equivariant $(\infty, 1)$-categories

TL;DR

This paper demonstrates that existing models for $( abla, 1)$-categories can be extended to include equivariant versions for any discrete group, and in some cases, for simplicial groups, broadening their applicability.

Contribution

It introduces equivariant extensions of known models for $( abla, 1)$-categories, including actions of simplicial groups, which was not previously established.

Findings

All known models extend to equivariant versions for any discrete group.

Two models support actions of simplicial groups.

The extensions preserve the models' core properties.

Abstract

In this paper we show that the known models for $(\infty, 1)$-categories can all be extended to equivariant versions for any discrete group $G$. We show that in two of the models we can also consider actions of any simplicial group $G$.