Swiss-Cheese operad and Drinfeld center

TL;DR

This paper constructs a groupoid-based model for the Swiss-Cheese operad, relating it to classical algebraic descriptions, extending it rationally, and comparing it to formality-based models.

Contribution

It introduces a novel groupoid model for the Swiss-Cheese operad and explores its rational extension and formality comparison.

Findings

Established a groupoid model using parenthesized permutations and braids.

Connected the model to classical algebraic descriptions of the operad.

Compared the rational model to the formal operad model.

Abstract

We build a model in groupoids for the Swiss-Cheese operad, based on parenthesized permutations and braids, and we relate algebras over this model to the classical description of algebras over the homology of the Swiss-Cheese operad. We extend our model to a rational model for the Swiss-Cheese operad, and we compare it to the model that we would get if the operad Swiss-Cheese were formal.