$A_\infty$-actions and Recognition of Relative Loop Spaces

TL;DR

This paper demonstrates that relative loop spaces can be characterized by $A_ abla$-actions, utilizing a 2-sided bar construction and operad homotopy equivalences to establish recognition theorems.

Contribution

It introduces the operad ${ m Act}_ abla$ of $A_ abla$-actions and shows its homotopy equivalence with the Swiss-cheese operad, advancing the understanding of relative loop space recognition.

Findings

Recognition of relative loop spaces via $A_ abla$-actions.

Construction of the operad ${ m Act}_ abla$ using Boardman-Vogt resolution.

Establishment of homotopy equivalence with the Swiss-cheese operad.

Abstract

We show that relative loop spaces are recognized by $A_\infty$-actions. A certain version of the 2-sided bar construction is used to prove such recognition theorem. The operad ${\sf Act}_\infty$ of $A_\infty$-actions is presented in terms of the Boardman-Vogt resolution of the operad ${\sf Act}$. We exhibit an operad homotopy equivalence between such resolution and the $1$-dimensional Swiss-cheese operad $\mathcal{SC}_1$.