nLab
model structure on dg-categories

Idea

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

Statement

Theorem

Let $k$ be a commutative ring. Write ${\mathrm{dgCat}}_k$ for the category of small dg-categories over $k$.

There is the structure of a cofibrantly generated model category on ${\mathrm{dgCat}}_k$ where a dg-functor $F:A\to B$ is

• a weak equivalence if

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

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

• a fibration if

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

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

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.

this model structure should be a presentation of the (∞,1)-category of stable (∞,1)-categories over the ring $k$.

Related concepts

• model structure on dg-operads

• model structure on dg-algebras over an operad

  • model structure on dg-algebras

  • model structure on dg-categories

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

• Bernhard Keller, On differential graded categories (pdf)

There is also

• David Rosoff, Mapping spaces of $A_{\mathrm{\infty }}$-algebras (pdf)