model category

for ∞-groupoids

# 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 $k$ 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 $H^0(A)$.

###### 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 $A_\infty$-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

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