Descent theory and mapping spaces

Abstract

The purpose of this paper is to develop a theory of \((\infty , 1)\)-stacks, in the sense of Hirschowitz–Simpson’s ‘Descent Pour Les n–Champs’, using the language of quasi-category theory and the author’s local Joyal model structure. The main result is a characterization of \((\infty , 1)\)-stacks in terms of mapping space presheaves. An important special case of this theorem gives a sufficient condition for the presheaf of quasi-categories associated to a presheaf of model categories to be a higher stack. In the final section, we apply this result to construct the higher stack of unbounded complexes associated to a ringed site.