# Differential calculus of Hochschild pairs for infinity-categories

## TL;DR

This paper introduces a new conceptual construction for the algebraic calculus structure on Hochschild cohomology and homology spectra within infinity-categories, extending to equivariant cases.

## Contribution

It provides a novel operadic framework for Hochschild pairs in infinity-categories, generalizing previous structures and including equivariant contexts.

## Key findings

- Hochschild pairs admit a calculus-like algebraic structure
- The structure is encoded by a two-colored operad by Kontsevich and Soibelman
- Generalization to equivariant infinity-categories is achieved

## 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.