nLab
model structure on dg-categories

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Idea

A model category structure on the category of dg-categories that exhibits them as a presentation for stable (infinity,1)-categories.

Statement

With Dwyer-Kan weak equivalences

Theorem

Let kk be a commutative ring. Write dgCat kdgCat_k for the category of small dg-categories over kk.

There is the structure of a cofibrantly generated model category on dgCat kdgCat_k where a dg-functor F:ABF : A \to B is

  • a weak equivalence if

    1. for all objects x,yAx,y \in A the component F x,y:A(x,y)B(F(x),F(y))F_{x,y} : A(x,y) \to B(F(x), F(y)) is a quasi-isomorphism of chain complexes;

    2. the induced functor on homotopy categories H 0(F)H^0(F) (obtained by taking degree 0 chain homology in each hom-object) is an equivalence of categories.

  • a fibration if

    1. for all objects x,yAx,y \in A the component F x,yF_{x,y} is a degreewise surjection of chain complexes;

    2. for each isomorphism F(x)ZF(x) \to Z in H 0(B)H^0(B) there is a lift to an isomorphism in H 0(A)H^0(A).

This is due to (Tabuada).

Remark

The definition is entirely analogous to the model structure on sSet-categories. Both are special cases of the model structure on enriched categories.

With Morita equivalences

There is another model category structure with more weak equivalences, the Morita equaivalences (Tabuada 05).

That is a presentation of the (∞,1)-category of stable (∞,1)-categories (Cohn 13).

References

The model structure on dg-categories is due to

  • Gonçalo Tabuada, Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories C. R. Acad. Sci. Paris Sér. I Math. 340 (1) (2005), 15–19.

It is reproduced as theorem 4.1 in

Discussion of internal homs of dg-categories in terms of refined Fourier-Mukai transforms is in

  • Bertrand Toën, The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007), 615–667

Discussion of internal homs of dg-categories using (just) the structure of a category of fibrant objects is in

  • Alberto Canonaco, Paolo Stellari, Internal Homs via extensions of dg functors (arXiv:1312.5619)

There is also

  • David Rosoff, Mapping spaces of A A_\infty-algebras (pdf)

The model structure with Morita equivalences as weak equivalences is discussed in

  • Goncalo Tabuada, Invariants additifs de dg-catgories. Internat. Math. Res. Notices 53 (2005), 33093339.

That the Morita model structure on dg-categories presents the homotopy theory of kk-linear stable (infinity,1)-categories was shown in

Revised on March 20, 2014 03:31:27 by Urs Schreiber (89.204.138.150)