The operad associated to a crossed simplicial group

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.