Homotopy theory of dg sheaves

TL;DR

This paper investigates the homotopy theory of differential graded sheaves by analyzing the local projective model structure on presheaves of complexes, characterizing fibrant objects via descent conditions and providing explicit replacement functors.

Contribution

It characterizes fibrant objects as those satisfying descent with respect to hypercovers and constructs cofibrant and fibrant replacement functors with desirable properties.

Findings

Fibrant objects satisfy descent with respect to all hypercovers

Explicit descriptions of cofibrant and fibrant replacement functors

Clarifies the local projective model structure on presheaves of complexes

Abstract

In this note we study the local projective model structure on presheaves of complexes on a site, i.e. we describe its classes of cofibrations, fibrations and weak equivalences. In particular, we prove that the fibrant objects are those satisfying descent with respect to all hypercovers. We also describe cofibrant and fibrant replacement functors with pleasant properties.