Homology, Homotopy and Applications

Volume 19 (2017)

Number 2

Explicit homotopy limits of $\mathrm{dg}$-categories and twisted complexes

Pages: 343 – 371

DOI: https://dx.doi.org/10.4310/HHA.2017.v19.n2.a17


Jonathan Block (Department of Mathematics, University of Pennsylvania, Philadelphia, Penn., U.S.A.)

Julian Holstein (Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster, United Kingdom)

Zhaoting Wei (Department of Mathematical Sciences, Kent State University, Kent, Ohio, U.S.A.)


In this paper we study the homotopy limits of cosimplicial diagrams of $\mathrm{dg}$-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.


differential graded category, twisted complex

2010 Mathematics Subject Classification

14F05, 18D20, 18G55

The second author was supported by EPSRC grant EP/N015452/1 for part of this work.

Received 30 November 2015

Received revised 27 April 2017

Published 29 November 2017