Tangent categories of algebras over operads

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