Interaction of Grothendieck Construction with Coherent Nerve

Question

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?

Answer 1

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!