Explicit homotopy limits of $\mathrm{dg}$-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 O-modules on theČech nerve of an open cover of a ringed space (X, 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.

Homotopy limits in the category of dg-categories in terms of $$ \mathrm {A}_{\infty } $$-comodules

“…Much of the inspiration comes from Block, Holstein, and Wei (2017). In the process, we prove their Conjecture 1 and generalize their results.…”

Gluing semi-orthogonal decompositions

“…• Since D l is simple normal crossing, for every l we can equip Perf( r (U l , D| U l )) with the psod of type Z D l ,r given by Proposition 4.1, where the semi-orthogonal factors are defined as in (6). Also, since Z D l ,r = Z D,r , we can write Perf( r (U l , D l )) = A l,k χ , χ ∈ (Z D,r , ≤) .…”

Twisted complexes on a ringed space as a dg-enhancement of the derived category of perfect complexes

“…4 we talk about some applications of twisted complexes. In particular we illustrate the application in descent theory.…”

“…This problem has been solved in the recent preprint [4] and the construction depends heavily on the simplicial resolution of dg-categories in [10]. …”

“…Remember that the map is the dot multiplication of a 0,1 i as in (4). Now we consider the map a 1,0 ii : E n i → E n i , the Maurer-Cartan equation (7) in the k = 1 case tells us…”