Glueing a property via homotopy colimits

Question

I have a problem concerning a fact which is stated without proof in this Rezk's draft: http://www.math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf . In the proof of Lemma 2.11, we are given a fibration $p:E \to B$ and a diagram $$V: J \to Psh(I)/B$$ where $Psh(I)$ denotes simplicial presheaves on $I$, such that any arrow $V(j) \to B$ is such that the pullback of $p$ along it gives us a square (a homotopy cartesian one, of course) which remains homotopy cartesian after having applied the homotopy colimit functor $$hocolim:Psh(I)\to sSet$$ The claim is that the map $$hocolim(V)\to B$$ should then have the same property. What I have done so far is firstly having interpreted this claim as: pick a cofibrant replacement $\mathcal{Q}V\simeq V$ in $Psh(I)^J$ and consider the map $colim \mathcal{Q}V \to B$, we want it to have the abovementioned property (notice that the maps $\mathcal{Q}V(j) \to B$ have it, since they are weakly equivalent to the $V(j) \to B$ which have it by assumption).

I would like to use the following result, but I haven't find a way in which it could turn out to solve my problem:

Here, $E \to B$ is equifibered if the following square is homotopy cartesian for any choice of $J_1,J_2,\alpha$:

And $E \to B$ is a realization-fibration if any homotopy cartesian square of the form remains homotopy cartesian after having applied the homotopy colimit functor

Thanks in advance for any hint or advice.

Answer 1

I don't think you want to apply 2.6 here. This is supposed to follow from "descent", or more precisely, from the fact that hocolims are stable under base change.

Let me write $|F|$ for $\mathrm{hocolim}_{I^\mathrm{op}} F$, where $F$ is a presheaf on $I$. Given the $J$-cone in $I$-presheaves $V$, define $W(j):= V(j)\times_B E$. The hypothesis is that for each object $j$ of $J$, the map $$ |W(j)| \to \mathrm{holim} \bigl( |V(j)| \to |B| \leftarrow |E| \bigr) $$ is a weak equivalence of spaces.

I want to show that $$\def\hocolim{\mathrm{hocolim}} |\hocolim_J W| \to \mathrm{holim}\bigl( |\hocolim_J V| \to |B| \leftarrow |E|\bigr). $$ As you note, it is convenient to model $V$ by a projective cofibrant functor (with respect to the $J$-variable), so that $\hocolim_J V=\mathrm{colim}_J V$.

The claim follows if I can show that the above map is equivalent to $$ \hocolim_J|W(-)|\to \hocolim_J(\mathrm{holim}\bigl(|V(-)| \to |B| \leftarrow |E|\bigr) $$ since this map is induced by applying $\hocolim$ to a $J$-levelwise weak equivalence. On the domain, this is just the fact that $\hocolim_J$ commutes with $|-|=\hocolim_I$. For the codomain, we need to add the fact that homotopy colimits are compatible with homotopy base change, i.e., the evident map $$ \hocolim_J(\mathrm{holim}\bigl( |V(-)| \to |B| \leftarrow |E|\bigr) \xrightarrow{\sim} \mathrm{holim}\bigl( \hocolim_J|V(-)| \to |B| \leftarrow |E|\bigr) $$ is a weak equivalence.