nLab
model structure on enriched categories

Idea

If $V$ is a monoidal model category, then in many cases there is a model category of $V$-enriched categories. This includes the model structure on simplicial categories and the model structure on dg-categories, for instance.

Definition

Let $V$ be a monoidal model category. The localization functor $\gamma :V\to \mathrm{Ho}(V)$ is then a lax monoidal functor, and hence any $V$-category $C$ induces a $\mathrm{Ho}(V)$-category ${\gamma }_{•}C$. The homotopy category of a $V$-category $C$ is the underlying ordinary category $({\gamma }_{•}C{)}_o$. We say a $V$-functor $F:C\to D$ is locally X if each morphism $F:C(x,y)\to D(Fx,Fy)$ is X.

Define a $V$-functor $F:C\to D$ to be:

• A weak equivalence if ${\gamma }_{•}F:{\gamma }_{•}C\to {\gamma }_{•}D$ is an equivalence of $\mathrm{Ho}(V)$-categories (that is, an internal equivalence in the 2-category of $\mathrm{Ho}(V)$-categories). This is equivalent to asking that (1) $F$ is locally a weak equivalence, and (2) the ordinary functor $({\gamma }_{•}F{)}_o$ is essentially surjective.

• A naive fibration if (1) $F$ is locally a fibration, and (2) ${\gamma }_{•}F$ is an isofibration.

Define a $V$-category $C$ to be

• fibrant if the functor $C\to 1$ is a naive fibration. This is equivalent to $C$ being locally fibrant, i.e. each $C(x,y)$ is fibrant.

By a theorem of Joyal, these weak equivalences and fibrant objects determine at most one model structure on the category $V\mathrm{Cat}$. When it exists, it is called the (canonical, categorical) model structure on $V$-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 $V$-functors that are (1) locally trivial fibrations and (2) surjective on objects.

See the references for general conditions under which this model structure exists.

Examples

• The model structure on simplicial categories which presents (∞,1)-categories is induced from the Quillen model structure on simplicial sets.

• The model structure on simplicial categories which presents (∞,2)-categories is induced from the Joyal model structure on simplicial sets

• The model structure on dg-categories is induced from the projective model structure on chain complexes.

• The canonical model structure on Cat is induced from the trivial model structure on Set.

• The canonical (Lack) model structure on $2\mathrm{Cat}$ is induced from the canonical model structure on $\mathrm{Cat}$.

• The canonical (Lack) model structure on Gray-categories is induced from the canonical model structure on $2\mathrm{Cat}$.

References

• Jacob Lurie, Higher Topos Theory section A.3.2

• Alexandru Stanculescu?, “Constructing model categories with prescribed fibrant objects”, arXiv