Tangent categories of algebras over operads

TL;DR

This paper develops a model-categorical framework for tangent categories of algebras over operads, extending existing results to enriched operads in non-stable model categories, with applications to cotangent complexes.

Contribution

It extends the identification of tangent categories to enriched operads in non-stable model categories, broadening the scope of operadic algebra analysis.

Findings

Established a model-categorical counterpart for tangent categories of operad algebras.

Extended the comparison to enriched operads in non-stable model categories.

Provided tools for computing cotangent complexes of enriched categories.

Abstract

Associated to a presentable $\infty$-category $\mathcal{C}$ and an object $X \in \mathcal{C}$ is the tangent $\infty$-category $\mathcal{T}_X\mathcal{C}$, consisting of parameterized spectrum objects over $X$. This gives rise to a cohomology theory, called Quillen cohomology, whose category of coefficients is $\mathcal{T}_X\mathcal{C}$. When $\mathcal{C}$ consists of algebras over a nice $\infty$-operad in a stable $\infty$-category, $\mathcal{T}_X\mathcal{C}$ is equivalent to the $\infty$-category of operadic modules, by work of Basterra--Mandell, Schwede and Lurie. In this paper we develop the model-categorical counterpart of this identification and extend it to the case of algebras over an enriched operad, taking values in a model category which is not necessarily stable. This extended comparison can be used, for example, to identify the cotangent complex of enriched categories, an application we take up in a subsequent paper.