TY - JOUR

T1 - Explicit homotopy limits of dg-categories and twisted complexes

AU - Block, Jonathan

AU - Holstein, Julian V. S.

AU - Wei, Zhaoting

N1 - ©2017 by International Press of Boston, Inc. All rights reserved.

PY - 2017/11/29

Y1 - 2017/11/29

N2 - 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.

AB - 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.

KW - math.CT

KW - math.AG

KW - math.AT

KW - 18D20, 18G55, 14F05

KW - differential graded category

KW - twisted complex

U2 - 10.4310/HHA.2017.v19.n2.a17

DO - 10.4310/HHA.2017.v19.n2.a17

M3 - Journal article

VL - 19

SP - 343

EP - 371

JO - Homology, Homotopy and Applications

JF - Homology, Homotopy and Applications

SN - 1532-0073

IS - 2

ER -