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Abstract. We develop the global period map in the context of derived geometry, generalising Griffiths' classical period map as well as the infinitesimal derived period map. We begin by constructing the derived period domain which classifies Hodge filtrations and enhances the classical period domain. We analyze the monodromy action. Then we associate to any polarized smooth projective map of derived stacks a canonical morphism of derived analytic stacks from the base into the quotient of the derived period domain by monodromy. We conclude the paper by discussing a few examples and a derived Torelli problem. In the appendix we describe how to present derived analytic Artin stacks as hypergroupoids, which may be of independent interest., Comment: 66 pages; removed incorrect statement from appendix and added explicit check that derived period map is well-defined

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., Comment: 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

We identify Morita cohomology, which is a categorification of the cohmology of a topological space X, with the category of homotopy locally constant sheaves of perfect complexes on X., Comment: 12 pages

We consider two categorifications of the cohomology of a topological space X by taking coefficients in the category of differential graded categories. We consider both derived global sections of a constant presheaf and singular cohomology and find the resulting dg-categories are quasi-equivalent and moreover quasi-equivalent to representations in perfect complexes of chains on the loop space of X., Comment: 33 pages

We provide proofs of two properties of the model category dgCat of dg-categories (with the Morita or Dwyer-Kan model structure): When working over a field the category dgCat is left proper. Natural simplicial resolutions in dgCat are given by dg-categories of Maurer-Cartan elements., Comment: 15 pages. V5: added appendix to close two gaps in the proof of the main theorem. V3 appeared in Homology, Homotopy and Applications, vol. 16(2), 2014