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

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 $\mathsf{E}_n$ algebra), $B$ is an $\mathsf{E}_{n-1}$-algebra, and $f : A \to Z(B)$ 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 $A \otimes B \to B$; in the extended Swiss-Cheese operad $\mathsf{ESC}_{mn}$, the lower algebra is an $\mathsf{E}_m$-algebra for some $m < n$; and in the complementarily-constrained disks operad $\mathsf{CD}_{mn}$, the morphism is replaced by a derivation $f + \epsilon \delta : A \to B[\epsilon]$. 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.