Swiss-cheese action on the totalization of operads under the monoid actions actions operad

TL;DR

This paper demonstrates that semi-totalizations of semi-cosimplicial spaces derived from colored operads have homotopy types related to double loop spaces and are equivalent to algebras over the Swiss-cheese operad, revealing new connections in operad theory.

Contribution

It establishes a link between semi-totalizations of semi-cosimplicial spaces from colored operads and their homotopy types as double loop spaces and Swiss-cheese operad algebras.

Findings

sTot(Y) has the homotopy type of a relative double loop space

(sTot(X), sTot(Y)) is weakly equivalent to an algebra over the Swiss-cheese operad

Provides a new perspective on the structure of operad-derived spaces

Abstract

We prove that if a pair of semi-cosimplicial spaces (X,Y) arise from a coloured operad then the semi-totalization sTot(Y) has the homotopy type of a relative double loop space and the pair (sTot(X),sTot(Y)) is weakly equivalent to an explicit algebra over the two dimensional Swiss-cheese operad.