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Created model structure on enriched categories. I’m surprised we didn’t already have this (unless it’s under some other name I didn’t find).
Yes, true, I guess we didn’t have this yet. Thanks for creating it.
Does the model structure on $Cat$-enriched categories induced by enrichment in the canonical model structure on $Cat$ give the correct $(\infty, 1)$-category of $(2,2)$-categories? (or maybe the correct $(\infty,2)$-category or $(\infty,3)$-category of $(2,2)$-categories? I’m getting mixed messages on what precisely is supposed to be presented by enrichment in model categories)
I’ve been flipping through the various nLab pages, and I can’t find a clear statement, and this is further confounded by what seems to be inconsistent usage of $2Cat$.
Re #3: Have you seen https://arxiv.org/abs/1312.3881? As far as I remember, it answers your question.
The following statement on the page would imply that it gives the correct $(\infty,1)$-category of $(2,2)$-categories.
The canonical (Lack) model structure on 2Cat is induced from the canonical model structure on Cat.
Since everything is fibrant, this amounts to checking that the weak equivalences are correct, which does seem to be the case, i.e. the description of the weak equivalences at model structure on enriched categories in this case seems to be the following one at equivalence of 2-categories:
A 2-functor can be made into part of an equivalence iff it is essentially surjective on objects, essentially full on 1-cells (i.e. essentially surjective on Hom-categories), and fully faithful on 2-cells.
One thing that confused me was that canonical model structure on 2-categories says the weak equivalences in Lack’s model structure are the ones that also have a strict inverse (up to strict natural isomorphism). But now I see Lack’s paper actually describes what you do.
But the other thing that was bothering me is about having enough functors. I’d been trying to find a statement a coherence theorem that said any functor between 2-categories can be transported along equivalences to a strict functor between strict 2-categories, but the only statements I could find were either just about the objects, or only made a statement involving strict 2-categories and pseudofunctors (and natural transformations and modifications).
Re #7: The description on the nlab page to which you link of the weak equivalences in Lack’s model structure on 2-Cat as the “strict equivalences” is incorrect.
I’m not quite sure what you’re asking in your second paragraph. One relevant fact is that for any cofibrant 2-category $A$, any 2-category $B$, and any pseudofunctor $F \colon A \to B$, there exists a strict 2-functor $G \colon A \to B$ and an invertible icon $F \cong G$. But probably more what you are after is the fact that for any pseudofunctor between bicategories $F \colon A \to B$, there exists a strict 2-functor between strict 2-categories $G \colon C \to D$ and bijective-on-objects biequivalence pseudofunctors $U \colon A \to C$ and $V \colon B \to D$ such that $V F = G U$; see e.g. §2.3.3 of Nick Gurski’s book on Coherence in three-dimensional category theory.
Re #7: The description on the nlab page to which you link of the weak equivalences in Lack’s model structure on 2-Cat as the “strict equivalences” is incorrect
I think that what was intended on that page was to refer to the fact that the functors are strict. I’ll correct it.
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