related by the Dold-Kan correspondence
If is a monoidal model category, then in many cases there is a model category of -enriched categories. This includes the model structure on simplicial categories and the model structure on dg-categories, for instance.
Let be a monoidal model category. The localization functor is then a lax monoidal functor, and hence any -category induces a -category . The homotopy category of a -category is the underlying ordinary category . We say a -functor is locally X if each morphism is X.
Define a -functor to be:
A weak equivalence if is an equivalence of -categories (that is, an internal equivalence in the 2-category of -categories). This is equivalent to asking that (1) is locally a weak equivalence, and (2) the ordinary functor is essentially surjective.
A naive fibration if (1) is locally a fibration, and (2) is an isofibration.
Define a -category to be
By a theorem of Joyal, these weak equivalences and fibrant objects determine at most one model structure on the category . When it exists, it is called the (canonical, categorical) model structure on -categories.
Usually, the fibrations between fibrant objects in this model structure are precisely the naive fibrations (although between non-fibrant objects, the two classes are distinct). Usually also, the trivial fibrations are precisely the weak equivalences that are also naive fibrations, which is to say the -functors that are (1) locally trivial fibrations and (2) surjective on objects.
See the references for general conditions under which this model structure exists.
The canonical (Lack) model structure on is induced from the canonical model structure on .
The canonical (Lack) model structure on Gray-categories is induced from the canonical model structure on .