Twisted tensor product of dg categories and Kontsevich's Swiss Cheese conjecture
Michael Batanin, Boris Shoikhet

TL;DR
This paper constructs a dg operad model for the Swiss Cheese operad, proving a chain-level equivalence of action categories that confirms a version of Kontsevich's Swiss Cheese conjecture.
Contribution
It introduces a colored dg operad model for the Swiss Cheese operad, establishing a chain-level equivalence that verifies the conjecture in a stricter form.
Findings
Constructed a colored dg operad ${O}$ weakly equivalent to the Swiss Cheese operad.
Proved an equivalence of categories between algebra actions and Hochschild complexes.
Validated the conjecture at the chain level without passing to homotopy categories.
Abstract
Let be a -algebra. The Kontsevich Swiss Cheese conjecture [K2] states that the homotopy category of actions of -algebras on has a final object and that this object is weakly equivalent to the pair where is the Hochschild complex of . Here the category of actions is the category whose objects are pairs which are algebras of the chain Swiss Cheese operad such that the induced action of the little interval operad on the component coincides with the -structure on . We prove that there is a colored dg operad with 2 colors, weakly equivalent to the chain Swiss Cheese operad for which the following ``stricter" version of the Kontsevich Swiss Cheese conjecture holds. Denote the two colors of by (for the 1-algebra argument) and (for the 2-algebra argument), denote…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology · Advanced Algebra and Geometry
