nLab model structure on dg-categories

Idea

The general notion model structure on enriched categories gives in particular a model structure on dg-categories, called the Dwyer-Kan model structure, and analogous to the usual model structure on sSet-categories which models (infinity,1)-categories.

There are interesting left Bousfield localizations of this model structure, called the quasi-equiconic and Morita model structures. Here the fibrant objects are the pretriangulated dg-categories, resp. idempotent complete pretriangulated dg-categories. In characteristic zero, the Morita model structure is known to present the (infinity,1)-category of linear stable (infinity,1)-categories (Cohn 13).

Statement

With Dwyer-Kan equivalences

Theorem

Let $k$ be a commutative ring. Write $dgCat_k$ for the category of small dg-categories over $k$.

There is the structure of a cofibrantly generated model category on $dgCat_k$ 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.

With Morita equivalences

There is another model category structure with more weak equivalences, the Morita equivalences (Tabuada 05). This is in fact the left Bousfield localization of the above model structure with respect to the Morita equivalences, i.e. functors $F: C \to D$ whose induced restriction of scalars functor $\mathbf Lf^* : \mathbf D(D) \to \mathbf D(C)$ is an equivalence of categories.

The fibrant objects with respect to this model structure are the dg-categories A for which the canonical inclusion $H^0(A) \hookrightarrow \mathbf D(A)$ has its essential image stable under cones, suspensions, and direct sums. Hence the homotopy category with respect to this model structure is identified with the full subcategory of Ho(DGCat), the homotopy category of the Dwyer-Kan model structure, spanned by dg-categories of this form.

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

The pretriangulated envelope of Bondal-Kapranov is a fibrant replacement functor for the Morita model structure. The DG quotient? of Drinfeld is a model for the homotopy cofibre with respect to the Morita model structure.

Quasi-equiconic model structure

Here the fibrant objects are the pretriangulated dg-categories. (…)

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)

A summary of the various model structures on dg-categories:

• Denis-Charles Cisinski, Goncalo Tabuada, Section 2 of: Non-connective K-theory via universal invariants, Compositio Math. 147 (2011) 1281-1320 [arXiv:0903.3717, pdf]

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)

The derived internal Hom in the homotopy category of DG-categories is equivalent to the dg-category of A_infty-functors.

• Giovanni Faonte, A-infinity functors and homotopy theory of DG-categories, arXiv.

A proof that the internal hom of Ho(DGCat) constructed by Toën is in fact the right derived functor of the internal hom of DGCat is in

• Beatriz Rodriguez Gonzalez, A derivability criterion based on the existence of adjunctions, 2012, arXiv:1202.3359.

There is also

• David Rosoff, Mapping spaces of $A_\infty$-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 $k$-linear stable (infinity,1)-categories was shown in

• Lee Cohn, Differential Graded Categories are k-linear Stable Infinity Categories (arXiv:1308.2587)

See also

• Piergiorgio Panero, Boris Shoikhet, A Quillen model structure on the category of Kontsevich-Soibelman weakly unital dg categories (arXiv:1907.07970)

• Piergiorgio Panero, Boris Shoikhet, A closed model structure on the category of weakly unital dg categories, II (https://arxiv.org/abs/2007.12716)