On D-modules of categories II

TL;DR

This paper develops two methods for constructing D-module structures on periodic cyclic homology of stable infinity categories and proves their equivalence, advancing the understanding of categorical D-modules.

Contribution

It introduces and compares two novel approaches for constructing D-module structures on periodic cyclic homology in the context of stable infinity categories.

Findings

The two methods produce equivalent D-module structures.

The canonical extension of factorization homology is effective.

The relationship between Hochschild (co)homology and D-modules is clarified.

Abstract

In our paper "On D-module of categories I", we provide two different methods of constructing D-module structures on the complex computing periodic cyclic homology associated to a family of stable infinity categories. One is based on a canonical extension of factorization homology. Another method uses the pair of Hochschild cohomology and Hochschild homology, Kodaira-Spencer map for a family of stable infinity categories, Koszul dualities,and the relation between dg Lie algebras and pointed formal stacks. In this paper, we prove that two resulting structures coindice.