The homotopy theory of dg-categories and derived Morita theory

By Bertrand Toën. 2004

564 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bf5b954aa1f5dd7dd22dc0629a86353246b490d4

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Citations in the corpus

On triangulated orbit categories (2005, background): “These include the category of projectivem odulesovera preprojectivealgebra ofgeneralized D ynkin typein thesenseof Happel-Preiser-Ringel[29],whosetriangulated structuregoesback to Auslander-Reiten’s work [6],[44],[7].”
A tour about existence and uniqueness of dg enhancements and lifts (2016, background): “The tensor product in dgCat can be appropriately derived (see [31] or [65]).”
Triangulated surfaces in triangulated categories (2013, methodology): “By the main result of [Toë07], the dg-category RHom(A,B) can be explicitly described as the full dg-subcategory of ModAop⊗B formed by those dg-modules M satisfying: (1) M is cofibrant.”
Symmetric monoidal structure on non-commutative motives (2010, methodology): “[3, 4, 16, 17, 33, 34, 35, 48], they are considered as dg-enhancements of derived categories of (quasi-)coherent sheaves on a hypothetic non-commutative space.”
Topological equivalences for differential graded algebras (2006, methodology): “A model category structure on dg-categories is constructed in [Ta], and later used in [Tö].”
Uniqueness of the multiplicative cyclotomic trace (2011, background): “However, the smash product of cofibrant spectral categories is not necessarily cofibrant, and consequently the model structure is not monoidal [30] (and see [34] for a discussion of this in the setting of DG-categories).”
Compact generators in categories of matrix factorizations (2009, methodology): “The existence of this model structure allows for a description of the mapping spaces in the category Ho(dgcatk[u,u−1]) as done for Ho(dgcatk) in [Toë07].”
Non-connective K-theory via universal invariants (2009, background): “The derived tensor monoidal structure −⊗L − on Hqe admits an internal Hom-functor (denoted by RHom(−,−) in [Toe07]) rep(−,−) : Hqeop × Hqe−→ Hqe.”
Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations (2010, methodology): “We work with Hochschild homology in the context of dg-categories following the framework developed by Toën in [32].”
Hopfological algebra (2012, methodology): “In subsequent works we will investigate this question in parallel with Toën’s framework [Toë07] on DG categories, as well as more related K-theoretical questions.”