2023/05/17 by Christopher Brav, Brav, Christopher, Nick Rozenblyum +1 · 2 citations
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2305.10323
openalex publication_date 2023/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Deligne conjecture (many times a theorem) endows Hochschild cochains of a linear category with the structure of an E2-algebra, that is, of an algebra over the little 2-disks operad. In this paper, we prove the cyclic Deligne conjecture, stating that for a linear category equipped with a Calabi-Yau structure (a kind of non-commutative orientation), the Hochschild cochains is endowed with the finer structure of a framed E2-algebra, that is, of a circle-equivariant algebra over the little 2-disks operad. Our approach applies simultaneously to both smooth and proper linear categories, as well as to linear functors equipped with a relative Calabi-Yau structure, and works for a very general notion of linear category, including any dualizable presentable ∞-category. As a particular application, given a compact oriented manifold with boundary ∂ M ⊂ M, our construction gives chain-level genus zero string topology operations on the relative loop homology H*(LM,L∂ M).