A combinatorial approach to Kontsevich's Swiss cheese conjecture

2025/12/23 by Florian De Leger, De Leger, Florian
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Advanced Topics in Algebra #Algebraic structures and combinatorial models

paper · doi:10.48550/arxiv.2512.20167

Abstract

From a coloured operad P and a P-algebra A, we construct a new operad SC(P) and a Hochschild object Hoch(A) together with an SC(P)-action on the pair (Hoch(A),A). We prove that if P is the little n-disks operad, then SC(P) is equivalent to the Swiss cheese operad SC_n. This gives us a weak version of Kontsevich's Swiss cheese conjecture (without the universal property). We apply our result to prove the existence of an E_n+1-action on the Hochschild-Pirashvili cochain of order n of a commutative algebra.

Citations

• Condensation of the operad for multiplicative hyperoperads
• Twisted tensor product of dg categories and Kontsevich's Swiss Cheese conjecture
• Quasi-tame substitudes and the Grothendieck construction
• Operads as polynomial 2-monads
• On Kontsevich's Hochschild cohomology conjecture
• Cosimplicial Objects and little n-cubes. I
• The Swiss-Cheese Operad

Related

• Twisted tensor product of dg categories and Kontsevich's Swiss Cheese conjecture
• From maps between coloured operads to Swiss-Cheese algebras
• The Swiss-Cheese Operad
• OCHA and the swiss-cheese operad
• (Non-)formality of the extended Swiss Cheese operads