12:08 AM
12:46 AM
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3:09 AM
@JonathanBeardsley Ok, so what this says is that the coherent nerve of the full subcategory of fibrant-cofibrant objects is categorically equivalent to the relative nerve of that simplicial model cat viewed as an ordinary model cat
Simplicial model categories are very special though. The whole point of using them is that to do homotopy-localization wrt the model structure, you only need to pass to full subcat of fibrant-cofibrant objects. You don't need any "localization technology"
Hinich is relating (marked) quasicats to relative categories to simplicial categories. The functor RelCat->SimpCat is the hammock localization, the functor SimpCat->MarkedSSet is the coherent nerve
4:05 AM
I guess, summing up what I was saying, thinking aloud, for simplicial model cats, this gives us five equivalent models of the localization as a quasicategory: The coherent nerve of the fibrant-cofibrant objects, the marked fibrant replacement of the relative simplicial nerve, the marked fibrant replacement of the marked relative nerve of the underlying model category, the relative nerve of the underlying model category, or the coherent nerve of the simplicial version of the hammock localization
If your simplicial cat is a simplicial model cat, I can't see any reason to look at the more complicated ones
The simplicial version of the DK localization is actually kind of strange. I think they are actually considering the simplicial cats as actual simplicial objects in Cat, and taking the derived functor of localization is done at each simplicial level. If I remember correctly, this is why D-K had a model structure on simplicially enriched categories all with the same set of objects.
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7:13 PM
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