Descent Theory and Mapping Spaces

TL;DR

This paper develops a theory of $( abla, 1)$-stacks using quasi-category theory, characterizing them via mapping space presheaves and applying the results to higher stacks of complexes.

Contribution

It introduces a characterization of $( abla, 1)$-stacks in terms of mapping space presheaves within the framework of quasi-categories.

Findings

Characterization of $( abla, 1)$-stacks using mapping space presheaves

A sufficient condition for presheaves of model categories to be higher stacks

Construction of the higher stack of unbounded complexes for a ringed site

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.