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There are a number of Grothendieck constructions: one for discrete categories, one for enriched categories (see Tamaki's paper here) and one for quasicategories (see the Unstraightening and Straightening correspondence of Lurie's Higher Topos Theory that goes between certain model categories whose underlying quasicategories are the ones of simplicial sets over a fixed simplicial set, and functors from that simplicial set to the quasicategory of quasicategories). Moreover, there is a construction that takes a simplicially enriched category whose morphism simplicial sets are Kan complexes and produces a quasicategory, namely the homotopy coherent nerve. My question is about how the coherent nerve construction interacts with the enriched Grothendieck construction and the Grothendieck construction on simplicial sets described by Lurie.

Now, suppose I have a pseudofunctor from a diagram category to simplicially enriched categories $F:I\to sCat$. The enriched Grothendieck construction gives me a functor $C\to I$, where $C$ is a simplicially enriched category. We can extend this functor along the homotopy coherent nerve $hN:sCat\to sSet$, and by Proposition 3.2.5.18 of Higher Topos Theory, we know that there is a corresponding morphism $X\to N(I)$ where $X$ is a simplicial set. How does this object relate to the coherent nerve applied to $C$, i.e. the morphism of simplicial sets $N(C)\to N(I)$, where $I$ is thought of as a simplicially enriched category by thinking of its morphism sets as 0-dimensional simplicial sets?

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  • $\begingroup$ Supposing that you can construct a natural functor $F:X \to N(C)$ or $F:N(C) \to X$ (over $N(I)$), you should be able to verify immediately that it is essentially surjective, and deduce from HTT 2.4.4.2 that it is fully-faithful. It might be easiest to construct $F$ using the relative nerve construction featured in HTT 3.2.5. $\endgroup$ Jun 19, 2016 at 21:02
  • $\begingroup$ Yeah, the main issue seems to be producing such a morphism. In my experience this is always the problem working with quasicategories... $\endgroup$ Jun 20, 2016 at 2:22
  • $\begingroup$ The technology of arxiv.org/abs/1605.00706 might help; I don't think it answers the question on its own, but it might provide a few links in a chain of Quillen equivalences that would. $\endgroup$ Jun 20, 2016 at 3:53
  • $\begingroup$ I think I managed to convince myself that if you construct X using the relative nerve construction of [HTT, Def. 3.2.5.2] then X is simply isomorphic as a simplicial set to $N(C)$. Is this ridiculously wrong? $\endgroup$ Jun 20, 2016 at 8:55
  • $\begingroup$ @YonatanHarpaz no I don't think this is wrong. In fact, I believe that this is true. I sort of started figuring out how the proof would go last night, but then I fell asleep. It's basically a matter of going through a tedious check that Hom(\Delta^n,X) is the same as Hom(\Delta^n,N(C)), I think... $\endgroup$ Jun 20, 2016 at 15:26

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Answering this question took me some time. First of all, Liang Ze Wong and I had to write down a version of the enriched Grothendieck construction that worked for these purposes, as Tamaki's construction, which I linked to above, didn't quite work. That paper can be found here.

Next, so long as we assume that $I$ is an ordinary category (it seems to take more work, thought it might be doable, in the case that $I$ is itself a simplicial category) then we can get that the simplicially enriched Grothendieck construction defined in the above cited paper, recovers Lurie's Grothendieck construction after applying the simplicial nerve.

So suppose we've got a functor $F:I\to sCat$. Then we can produce the functor $f:I\to sSet$ by composing with the simplicial nerve functor $N:sCat\to sSet.$ Then we can construct the relative nerve of $f$, denoted $N_f(I)$, from Chapter 3 of Lurie's HTT. Lurie shows then that there is a coCartesian fibration $N_f(I)\to N(I)$ and that this is the fibration associated to $f$ by his $\infty$-categorical Grothendieck construction. Then what Wong and myself show in this paper is that this relative nerve, $N_f(I)$ is actually isomorphic to the simplicial nerve of the enriched Grothendieck construction, $N(GrF)$ (this is Theorem 2.3.1 of this second cited paper). Since Lurie already told us that $N_f(I)\simeq Gr_\infty(f)$, we're done!

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  • $\begingroup$ Just realized that Yonatan Harpaz actually suggested this in a comment above, that I had forgotten about. Looks like we basically did the same thing! $\endgroup$ Aug 29, 2018 at 4:52

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