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


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?

• This is completely unrelated to your question, but that way to obtain a diagonal arrow in amscd is clever and painful at the same time. – fosco Dec 12 '20 at 13:37

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
in which they denote this left adjoint by $$r_!$$.
• Wow, I had no idea this construction went all the way back to the 80's. Also, this description of $\mathfrak{F}_{X}$ is lovely. Thank you very much! – Sofia Dec 8 '20 at 7:08