Tangent categories of algebras over operads

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.

arXiv commentThe section concerning stabilization of model categories was separated into an independent paper, appearing now as arXiv:1802.08031. Added an appendix on sifted homotopy colimits of algebras