# Colloque 2021 du GDR Topologie Algébrique et Applications @ Université de Strasbourg

## Abstract

The usual Swiss-Cheese operad encodes triplets $(A,B,f)$, where $A$ is an algebra over the little disks operad in dimension $n$ (i.e., an $E_{n}$ algebra), $B$ is an $E_{n−1}$-algebra, and $f:A→Z(B)$ is a central morphism of $E_{n}$-algebras. The Swiss-Cheese operad admits several variants and generalizations. In Voronov’s original version, the morphism is replaced by an action $A⊗B→B$; in the extended Swiss-Cheese operad $ESC_{mn}$, the lower algebra is an $E_{m}$-algebra for some $m<n$; and in the complementarily-constrained disks operad $CD_{mn}$, the morphism is replaced by a derivation $f+ϵδ:A→B[ϵ]$. In this talk, I will explain approaches to prove the (non-)formality of some of the variants of the Swiss-Cheese operad, including a joint work in progress with Renato Vasconcellos Vieira.