The Grothendieck construction is a classical correspondence between diagrams of categories and coCartesian fibrations over the indexing category. In this paper we consider the analogous correspondence in the setting of model categories. As a main result, we establish an equivalence between suitable diagrams of model categories indexed by M and a new notion of model fibrations over M. When M is a model category, our construction endows the Grothendieck construction with a model structure which gives a presentation of Lurie's ∞-categorical Grothendieck construction and enjoys several good formal properties. We apply our construction to various examples, yielding model structures on strict and weak group actions and on modules over algebra objects in suitable monoidal model categories.

Associated to a presentable ∞-category C and an object X ∈ C is the tangent ∞-category T X C, consisting of parameterized spectrum objects over X. This gives rise to a cohomology theory, called Quillen cohomology, whose category of coefficients is T X C. When C consists of algebras over a nice ∞-operad in a stable ∞-category, T X 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.

No abstract

In his fundamental work, Quillen developed the theory of the cotangent complex as a universal abelian derived invariant, and used it to define and study a canonical form of cohomology, encompassing many known cohomology theories. Additional cohomology theories, such as generalized cohomology of spaces and topological André-Quillen cohomology, can be accommodated by considering a spectral version of the cotangent complex. Recent work of Lurie established a comprehensive ∞-categorical analogue of the cotangent complex formalism using stabilization of ∞-categories. In this paper we study the spectral cotangent complex while working in Quillen's model-categorical setting. Our main result gives new and explicit computations of the cotangent complex and Quillen cohomology of enriched categories. For this we make an essential use of previous work, which identifies the tangent categories of operadic algebras in unstable model categories. In particular, we present the cotangent complex of an ∞-category as a spectrum valued functor on its twisted arrow category, and consider the associated obstruction theory in some examples of interest. Contents

No abstract