nLab
model structure on enriched categories

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Model structures on enriched categories

Idea

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

Definition

Let VV be a monoidal model category. The localization functor γ:VHo(V)\gamma: V \to Ho(V) is then a lax monoidal functor, and hence any VV-category CC induces a Ho(V)Ho(V)-category γ C\gamma_\bullet C. The homotopy category of a VV-category CC is the underlying ordinary category (γ C) o(\gamma_\bullet C)_o. We say a VV-functor F:CDF:C\to D is locally X if each morphism F:C(x,y)D(Fx,Fy)F:C(x,y) \to D(F x, F y) is X.

Define a VV-functor F:CDF:C\to D to be:

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

  • A naive fibration if (1) FF is locally a fibration, and (2) γ F\gamma_\bullet F is an isofibration.

Define a VV-category CC to be

  • fibrant if the functor C1C\to 1 is a naive fibration. This is equivalent to CC being locally fibrant, i.e. each C(x,y)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 VCatV Cat. When it exists, it is called the (canonical, categorical) model structure on VV-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 VV-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

References

Revised on April 10, 2014 03:37:48 by Urs Schreiber (145.116.131.80)