This is probably obvious... but given an equivalence of quasicategories, $F:C\to D$, do I get a (Cartesian) equivalence of marked simplicial sets $F^\natural:C^\natural\to D^\natural$?

Ah, yes, this is Proposition 3.1.3.5 of HTT.

Hey @HarryGindi and @RuneHaugseng, that theorem of Hinich sort of doesn't make sense to me. In particular... how does taking the nerve of a simplicial category, and the nerve of a wide subcategory, give me a marked simplicial set? It seems like it gives me a pair of simplicial sets...

Like, how do I apply that in practice to a, say, simplicial model category?

@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

In that particular case

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

and the functor RelCat->MarkedsSet is the relative nerve

If I read Hinich's lemma correctly, he says that the triangle of functors actually commutes

and you have to fibrantly replace after you do the hammock localization

Let me explain why it is nontrivial, I guess

He's not just doing a relative nerve on a fibrant relcat

He's doing a relative coherent nerve on a non-fibrant relative Simplicial cat

There isn't, afaik, a thomason model structure for simplicial cats, and no double subdivision trick

So this is a very nice result that says we can just take the relative coherent nerve and it is weakly equivalent to the thing we expect

It exhibits an explicit fibrant replacement

But yeah, for simplicial model categories, we could already do this easily

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

Some of these we discussed before

If your simplicial cat is a simplicial model cat, I can't see any reason to look at the more complicated ones

Two of them are just much more elaborate versions of the other two

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.

ICM 2018 highlights: ‘What did you enjoy? What did you learn?

Does anyone perhaps have an electronic copy of "Stable and Unstable Homotopy" Fields Institute Communications by AMS? As far as I can see, we don't have one at Northwestern library.