on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
A model category structure on the category of dg-categories that exhibits them as a presentation for stable (infinity,1)-categories.
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
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;
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
for all objects $x,y \in A$ the component $F_{x,y}$ is a degreewise surjection of chain complexes;
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).
The definition is entirely analogous to the model structure on sSet-categories. Both are special cases of the model structure on enriched categories.
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).
model structure on dg-algebras over an operad
model structure on dg-categories
The model structure on dg-categories is due to
It is reproduced as theorem 4.1 in
Discussion of internal homs of dg-categories in terms of refined Fourier-Mukai transforms is in
Discussion of internal homs of dg-categories using (just) the structure of a category of fibrant objects is in
There is also
The model structure with Morita equivalences as weak equivalences is discussed in
That the Morita model structure on dg-categories presents the homotopy theory of $k$-linear stable (infinity,1)-categories was shown in