The operad associated to a crossed simplicial group

TL;DR

This paper introduces operadic crossed simplicial groups, linking them to important operads like $E_$ and $E_2$, and demonstrates their applications in generalized bar constructions and homotopy theory.

Contribution

It defines operadic crossed simplicial groups, shows their relation to key operads, and applies this framework to generalized bar constructions and homotopy-theoretic spaces.

Findings

Symmetric and braid crossed simplicial groups can be made into operadic crossed simplicial groups.

The construction recovers the $E_$- and $E_2$-operads for symmetric and braid cases.

The framework generalizes bar constructions and relates to Baratt-Priddy-Quillen spaces.

Abstract

We introduce and study structured enhancement of the notion of a crossed simplicial group, which we call an operadic crossed simplicial group. We show that with each operadic crossed simplicial group one can associate a certain operad in groupoids. We demonstrate that symmetric and braid crossed simplicial groups can be made into operadic crossed simplicial groups in a natural way. For these two examples, we show that our construction of the associated operad recovers the $E_\infty$-operad and the $E_2$-operad respectively. We demonstrate the utility of this framework through two main applications: a generalized bar construction that specializes to Fiedorowicz's symmetric and braided bar constructions, and an identification of the associated group-completed monads with Baratt-Priddy-Quillen type spaces.