Deformations of quasi-categories in modules

TL;DR

This paper develops the deformation theory for quasi-categories in modules within the framework of templicial objects, showing their stability under certain infinitesimal deformations, thus advancing higher categorical concepts in enriched settings.

Contribution

It introduces the deformation theory for templicial modules and demonstrates the preservation of quasi-categories in modules under levelwise flat infinitesimal deformation.

Findings

Quasi-categories in modules are preserved under levelwise flat infinitesimal deformation.

The paper establishes foundational deformation results for templicial modules.

Advances understanding of higher categorical structures in enriched contexts.

Abstract

The framework of templicial objects was put forth in arXiv:2302.02484v1 in order to develop higher categorical concepts in the presence of enrichment. In particular, quasi-categories in modules constitute a subclass of templicial modules which may be considered as a kind of "weak dg-categories (concentrated in homologically positive degrees)" according to arXiv:2005.04778v3. The main goal of the present paper is to initiate the deformation theory of templicial modules. In particular, we show that quasi-categories in modules are preserved under levelwise flat infinitesimal deformation.