Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 097, 57 pages      arXiv:1904.02359      https://doi.org/10.3842/SIGMA.2020.097
Contribution to the Special Issue on Primitive Forms and Related Topics in honor of Kyoji Saito for his 77th birthday

Differential Calculus of Hochschild Pairs for Infinity-Categories

Isamu Iwanari
Mathematical Institute, Tohoku University, 6-3 Aramakiaza, Sendai, Miyagi, 980-8578, Japan

Received February 25, 2020, in final form September 04, 2020; Published online October 02, 2020

Abstract
In this paper, we provide a conceptual new construction of the algebraic structure on the pair of the Hochschild cohomology spectrum (cochain complex) and Hochschild homology spectrum, which is analogous to the structure of calculus on a manifold. This algebraic structure is encoded by a two-colored operad introduced by Kontsevich and Soibelman. We prove that for a stable idempotent-complete infinity-category, the pair of its Hochschild cohomology and homology spectra naturally admits the structure of algebra over the operad. Moreover, we prove a generalization to the equivariant context.

Key words: Hochschild cohomology; Hochschild homology; operad; $\infty$-category.

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