A combinatorial approach to Kontsevich’s Swiss cheese conjecture

The Swiss cheese operad was introduced and studied by Voronov [3]. It is a topological two-coloured operad, one colour corresponding to full disks and the other to half disks. The space of operations with target colour full disk is the space of ordered configuration of non-overlapping little disks inside the unit disk. The space of operations with target colour half disk is the space of ordered configuration of non-overlapping little disks or little half disks inside the unit half disk:

The operadic composition is given by plugging disks or half disks then reordering.

We will explain how to produce a new coloured operad $\mathrm{SC}(𝒫)$ from any coloured operad $𝒫$. We will then present the main result of [1] which states that if $𝒫$ is the little intervals operad, then $\mathrm{SC}(𝒫)$ is equivalent to the Swiss cheese operad. This gives us a weak version (without the universal property) of Kontsevich’s Swiss cheese conjecture [2].