Explicit homotopy limits of dg-categories and twisted complexes

TL;DR

This paper provides explicit constructions for homotopy limits of dg-categories, demonstrating their equivalence to totalizations in specific geometric and algebraic contexts, and introduces a stack of twisted perfect complexes.

Contribution

It offers an explicit model for homotopy limits of dg-categories using totalizations, applicable to sheaves on ringed spaces and group actions, and constructs a stack of twisted perfect complexes.

Findings

Explicit totalization models match homotopy limits in key cases.

Constructs a stack of twisted perfect complexes.

Provides models for twisted complexes and their variants.

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 \v{C}ech 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.