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:enter image description here

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

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

Thanks in advance for any hint or advice.


1 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.

  • $\begingroup$ Thanks a lot for your answer! I was thinking too that the result involved the descent properties of spaces, but I was also trying to follow the hint on the draft (maybe a typo?). $\endgroup$ Mar 30, 2015 at 18:56

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