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 $\gamma: V \to Ho(V)$ is then a lax monoidal functor, and hence any $V$-category $C$ induces a $Ho(V)$-category $\gamma_\bullet C$. The homotopy category of a $V$-category $C$ is the underlying ordinary category $(\gamma_\bullet C)_o$. We say a $V$-functor $F:C\to D$ is locally X if each morphism $F:C(x,y) \to D(F x, F y)$ is X.

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

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

• A naive fibration if (1) $F$ is locally a fibration, and (2) $\gamma_\bullet F$ 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 $V Cat$. 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.

Existence

Theorem

(Muro, [Muro], Theorem 1.1.) For any combinatorial closed symmetric monoidal model category $V$ satisfying the monoid axiom, the category $Cat(V)$ admits the Dwyer–Kan model structure. Moreover, this model structure is combinatorial.

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 $2 Cat$ is induced from the canonical model structure on $Cat$.

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

Related concepts

• model structure on internal categories

• model structure on operads

References

• Julia E. Bergner, A model category structure on the category of simplicial categories, Transactions of the American Mathematical Society 359:5 (2006), 2043-2058. arXiv, doi.

• Jacob Lurie, Higher Topos Theory section A.3.2

• Alexandru Stanculescu, Constructing model categories with prescribed fibrant objects, Theory and Applications of Categories, 29 23 (2014) 635-653 [arXiv:1208.6005, tac:29-23]

• Clemens Berger, Ieke Moerdijk, On the homotopy theory of enriched categories (arXiv:1201.2134)

• Fernando Muro, Dwyer–Kan homotopy theory of enriched categories. arXiv:1201.1575.