Explicit description of the left adjoint $\mathfrak{F}$ to the relative nerve

Question

$\newcommand{\op}{\mathrm{op}}\newcommand{\Un}{\mathrm{Un}}$Lurie's relative nerve functor $\mathrm{N}^{F}_{\bullet}$ is a simpler version of the unstraightening functor $$\Un_\phi:\mathrm{sPSh}(\mathcal{C}_\bullet)\to\mathsf{sSets}_{/S}$$ (or rather of its opposite; see HTT 3.2.5.1) which "unstraightens the homotopy coherence laws of a simplicial presheaf into a left fibration over $S_\bullet$" (I'm paraphrasing the nLab here)

Spelling out HTT 3.2.5.2, one sees that $\mathrm{N}^F_\bullet$ admits the following very simple description: it is the simplicial set where

• a vertex of $\mathrm{N}^F_\bullet$ is a pair $(A,x)$ with

  • $A$ an object of $\mathcal{C}$;
  • $x$ a vertex of $F(A)$;

• an edge of $\mathrm{N}^F_\bullet$ from $(A,x)$ to $(B,y)$ is a pair $(f,e)$ with

  • $f:A\to B$ a morphism of $\mathcal{C}$;
  • $e: F(f)(x)\to y$ an edge of $F(B)$;

• a $2$-simplex of $\mathrm{N}^F_\bullet$ is a $9$-tuple $(f,g,x,y,z,e_{01},e_{12},e_{02},\sigma)$ with

  • $f: A\to B$ a morphism of $\mathcal{C}$;
  • $g: B\to C$ a morphism of $\mathcal{C}$;
  • $x$ a vertex of $F(A)$;
  • $y$ a vertex of $F(B)$;
  • $z$ a vertex of $F(C)$;
  • $e_{01}: F(f)(x)\to y$ an edge of $F(B)$;
  • $e_{12}: F(g)(y)\to z$ an edge of $F(C)$;
  • $e_{02}: F(g\circ f)(x)\to z$ an edge of $F(C)$;
  • $\sigma: e_{12}\circ F(g)(e_{01})\Rightarrow e_{02}$ a $2$-simplex of $F(C)$ as in the diagram $$\require{AMScd} \require{cancel} \def\diaguparrow#1{\smash{\raise.6em\rlap{\!\!\!\!\!\!\scriptstyle #1} \lower.6em{\cancelto{}{\Space{2em}{1.7em}{0px}}}}} \begin{CD} && F(g)(y)\\ & \diaguparrow{F(g)(e_{01})} @VVe_{12}V \\ F(g\circ f)(x) @>>e_{02}> z \end{CD}$$

• and so on.

Now, in HTT 3.2.5.5, Lurie introduces a functor $\mathfrak{F}_X$ that is (FWIU) supposed to be a simpler analogue of the straightening functor $\mathrm{St}_\phi$. However, Lurie does so by invoking the adjoint functor theorem, similarly to the definition of $\Un_\phi$.

I gather that nevertheless those functors admit nice constructions (as shown for $\Un_\phi$ in Rezk's notes); so, my question is:

Question: What is a concrete description of Lurie's $\mathfrak{F}_X$ functor?

Answer 1

There is indeed a simple, explicit description of the left adjoint to the relative nerve functor. This left adjoint sends a simplicial set over the category $C$, i.e. a morphism of simplicial sets $p \colon X \to C$, to the functor $C \to \mathbf{sSet}$ that sends an object $c \in C$ to the simplicial set $X/c$ defined by the following pullback square of simplicial sets: $\require{AMScd}$ \begin{CD} X/c @>>> C/c\\ @V V V @VV \mathrm{dom} V\\ X @>>p> C \end{CD} A reference for this description of the left adjoint is the paper

Heuts, Gijs; Moerdijk, Ieke. Left fibrations and homotopy colimits. Math. Z. 279 (2015), no. 3-4, 723--744. doi

in which they denote this left adjoint by $r_!$.

Let me take this opportunity to remind everyone that the relative nerve construction appeared way back in the 1980's in §4 of the following paper written by the category theorist John Gray:

Gray, John W. The representation of limits, lax limits and homotopy limits as sections. Mathematical applications of category theory (Denver, Col., 1983), 63--83, Contemp. Math., 30, Amer. Math. Soc., Providence, RI, 1984. doi