Tangent categories of algebras over operads

Abstract

Associated to a presentable ∞-category $\mathcal{C}$ and an object X ∈ $\mathcal{C}$ is the tangent ∞-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 ∞-operad in a stable ∞-category, $\mathcal{T}_X\mathcal{C}$ is equivalent to the ∞-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.