The main purpose of this work is to study the homotopy theory of dg-categories up to quasi-equivalences. Our main result is a description of the mapping spaces between two dg-categories C and D in terms of the nerve of a certain category of (C, D)-bimodules. We also prove that the homotopy category Ho(dg −Cat) possesses internal Hom’s relative to the (derived) tensor product of dg-categories. We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dg-categories of modules over two dg-categories C and D as the dg-category of (C, D)-bi-modules. Finally, we give three applications of our results. The first one expresses Hochschild cohomology as endomorphisms of the identity functor, as well as higher homotopy groups of the classifying space of dgcategories (i.e. the nerve of the category of dg-categories and quasi-equivalences between them). The second application is the existence of a good theory of localization for dgcategories, defined in terms of a natural universal property. Our last application states that the dg-category of (continuous) morphisms between the dg-categories of quasi-coherent (resp. perfect) complexes on two schemes (resp. smooth and proper schemes) is quasi-equivalent

Abstract. Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, Dugger-Shipley,..., Toën and Toën-Vaquié. 1.

Abstract. We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly described in terms of a hypercover descent condition. These ideas are central to constructing realization functors on the homotopy theory of schemes [DI1, Is]. We give a few other applications for this new description of the homotopy theory of simplicial presheaves. Contents

Abstract not found

Abstract. We show that if two rings have equivalent derived categories then they have the same algebraic K-theory. Similar results are given for G-theory, and for a large class of abelian categories. Contents

Abstract not found

Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain complex of G-valued singular cochains on X. An alternative is to regard H n (•, G) as a representable functor on the homotopy category

Abstract. We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to Fulton-MacPherson and Axelrod-Singer in the setting of smooth manifolds, as well as a simplicial variant of this compactification. Our constructions are elementary and give simple global coordinates for the compactified configuration space of a general manifold embedded in Euclidean space. We stratify the canonical compactification, identifying the diffeomorphism types of the strata in terms of spaces of configurations in the tangent bundle, and give completely explicit local coordinates around the strata as needed to define a manifold with corners. We analyze the quotient map from the canonical to the simplicial compactification, showing it is a homotopy equivalence. We define projection maps and diagonal maps, which for the simplicial variant satisfy cosimplicial identities.

Stein manifolds, a prestack being a contravariant simplicial

Let Em denote the space of embeddings of the interval I = [−1, 1] in the cube I m with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy through immersions to the unknot – see Definition 5.1. By Proposition 5.17, Em is homotopy equivalent to Emb(I, I m) × ΩImm(I, I m). In [28], McClure and Smith define a cosimplicial object O • associated