Explicit homotopy limits of dg-categories and twisted complexes

Abstract

In this paper we study the homotopy limits of cosimplicial diagrams of dg-categories. We first give an explicit construction of the totalization of such a diagram and then show that the totalization agrees with the homotopy limit in the following two cases: (1) the complexes of sheaves of $\mathcal{O}$-modules on the Čech nerve of an open cover of a ringed space $(X,\mathcal{O})$; (2) the complexes of sheaves on the simplicial nerve of a discrete group $G$ acting on a space. The explicit models we obtain in this way are twisted complexes as well as their $D$-module and $G$-equivariant versions. As an application we show that there is a stack of twisted perfect complexes.

v3: 22 pages, minor changes, to appear in Homology, Homotopy and Applications. v2: a new subsection (Section 4.5) is added on the stack of twisted perfect complexes

References in corpus (3)

• Properness and simplicial resolutions for the model category dgCat
• Morita cohomology and homotopy locally constant sheaves
• Milnor descent for cohesive dg-categories

Cited by in corpus (2)

• Gluing semi-orthogonal decompositions
• Chern character for infinity vector bundles