Inventiones mathematicae

, Volume 167, Issue 3, pp 615–667 | Cite as

The homotopy theory of dg-categories and derived Morita theory

  • Bertrand ToënEmail author


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 dg-categories (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 dg-categories, 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 to the dg-category of quasi-coherent (resp. perfect) complexes on their product.


Model Category Isomorphism Class Natural Isomorphism Monoidal Category Homotopy Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Laboratoire Emile Picard UMR CNRS 5580Université Paul SabatierToulouse Cedex 9France

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