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I am trying to make sense of what is written in Rezk's draft http://www.math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf In particular, I am referring to Proposition 2.3, which is there stated without proof. According to different models for the homotopy colimit functor on $\text{sSet}^J$, we may or may not find maps $V_j \to \text{hocolim}_J V$ for any $j \in J$ and for any simplicial presheaf $V \in \text{sSet}^J$.

FIRST PROBLEM:In any case, it seems to me that it is impossible to get a strict cocone over $V$ with vertex $\text{hocolim}_J V$, as in the case of ordinary colimits. Hence, given a map $E \to \text{hocolim}_J V$ it makes no sense to speak of the functor $U(\cdot):J \to \text{sSet}$ defined by $U(\cdot):=V(\cdot) \times_{\text{hocolim}_J V} E$.

Is it possible that the author meant to use as a model the colimit of a cofibrant replacement $QV \to V$ in $\text{sSet}^J$, and so he is thinking of $QV_j \to \text{colim}_J QV\simeq \text{hocolim}_JV$ (thus committing an abuse of language)? In such case the definition of $U$ makes sense, and I think I can get a proof by using point (1) of Thm.1.4 in http://www.math.uiuc.edu/~rezk/rezk-sharp-maps.pdf .

SECOND PROBLEM: in the proof of point (1) of Thm.1.4 from the last link (page 14), I can't see why $\tilde{X}_i$ should be the pullback of $\text{colim}X'$ along the composite map $\tilde{Y}_i→ Y_i → \text{colim}Y$, and why it should imply that the right hand square is a pullback.

Thanks in advance for any help, which will of course be highly appreciated.

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  • $\begingroup$ It looks like the author is implicitly using the language of (infinity,1)-categories. $\endgroup$ – AAK Feb 15 '15 at 20:17
  • $\begingroup$ Yeah, I have been told so. Do you think it is possible to explain in full details the paper with the model categories language? And would you expand a bit your comment in an answer? I am really interested in $(\infty,1)$-stuff, but I am not by far expert enough to see what's going on. Thanks $\endgroup$ – Edoardo Lanari Feb 15 '15 at 20:19
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    $\begingroup$ I imagine it is possible to translate everything there to model categorical language, though I can't say for sure as I haven't read the notes. Doing so would require being careful about taking (co)fibrant replacements and so on. If you are not familiar with (infinity,1)-categorical language, you can probably just read the paper keeping in mind that you should take (co)fibrant replacements whenever necessary. $\endgroup$ – AAK Feb 15 '15 at 22:06
  • $\begingroup$ Hence you mean my guess is right? Is he (in your opinion) committing the abovementioned abuse of language? $\endgroup$ – Edoardo Lanari Feb 15 '15 at 22:08
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    $\begingroup$ Yes, but the point is that it is not an abuse of language in the world of (infinity,1)-categories, because there is no issue of (co)fibrancy there. $\endgroup$ – AAK Feb 15 '15 at 22:10
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For the first problem, as I wrote in the comments, the author is implicitly using the language of $(\infty,1)$-categories, where it does make sense to speak of such a functor. To understand why, you can consult any of the many introductions to $(\infty,1)$-category theory.

The rest of this answer deals with the second problem. $\newcommand{colim}{\mathop{\mathrm{colim}}} \newcommand{hocolim}{\mathop{\mathrm{hocolim}}}$ A fundamental property of any topos is universality of colimits, which means that colimits commute with base change: $$ (\colim_i X_i) \times_B A \simeq \colim_i (X_i \times_B A) $$ for any morphism $A \to B$. What Rezk calls the "distributive law" (Proposition 3.7) in his paper "Fibrations and homotopy colimits..." is just the special case of universality of colimits where $B = \colim_i X_i$.

In any "homotopy topos" or $\infty$-topos, there is the analogous property of commutativity of homotopy colimits with homotopy base change, i.e. $$ (\mathrm{hocolim}_i X_i) \mathop{\times}^h_B A \simeq \mathrm{hocolim}_i (X_i \mathop{\times}^h_B A) $$ for any morphism $A \to B$. In particular this holds in the archetypal example of a homotopy topos: the homotopy theory of simplicial sheaves. As a special case one has a "homotopy distributive law". Then Rezk's Theorem 1.4(1) in "Fibrations..." is a direct application of this homotopy distributive law. Indeed, in the notation of loc. cit., one sees $$\begin{align} \hocolim_i X_i &\simeq \hocolim_i (\colim X \mathop{\times}^h_{\colim Y} Y_i)\\ &\simeq \hocolim_i (\colim X \mathop{\times}^h_{\hocolim Y} Y_i) \\ &\simeq \mathrm{colim} X \end{align}$$ when $\colim Y \simeq \hocolim Y$. For a reference for this in the language of model categories, see paragraph 6.5 in these notes of Rezk (this is the property P1). In fact, the proof of the homotopy distributive law is the same as the proof of the ordinary distributive law as described by Rezk, using as a starting point the claim for spaces instead of for sets.

Since your question was rather about the proof in Rezk's older paper, let me try to say something about that. Let me emphasize that I have not really read the paper, so I may say something wrong in what follows. As far as I understood, the idea of the argument there is to use the distributive law for ordinary toposes, plus the explicit description of homotopy colimits of simplicial sheaves due to Bousfield-Kan.

For clarity, the proof can be divided into two steps: first, one demonstrates the claim assuming that $\colim X \to \colim Y$ is a sharp map. Then in general one reduces to the above case by factoring this map as a weak equivalence followed by a sharp map. The sharp version is:

Lemma: Let $Y : I \to s\mathscr{E}$ be a diagram and let $p : W \to \colim Y$ be a sharp map in $s\mathscr{E}$. Let $X : I \to s\mathscr{E}$ be the diagram defined by $X_i = Y_i \times_{\colim Y} W$ for each $i$. If $\hocolim Y \to \colim Y$ is a weak equivalence, then $\hocolim X \to \colim X$ is a weak equivalence.

Proof: First of all, the distributive law says there is an isomorphism $$ \colim_i X_i \approx \colim_i (Y_i \times_{\colim Y} W) \approx W. $$ Using the assumptions that $p$ is sharp and $Y$ is a homotopy colimit diagram, one gets the chain of weak equivalences $$\begin{align} \colim X &\simeq \colim X \mathop{\times}^h_{\colim Y} \hocolim Y \\ &\simeq \colim X \mathop{\times}_{\colim Y} \hocolim Y \\ &\simeq \colim X \mathop{\times}_{\colim Y} \colim \tilde{Y} \\ &\simeq \colim_i (\colim X \mathop{\times}_{\colim Y} \tilde{Y}_i) \\ &\simeq \colim_i \tilde{X}_i \\ &\simeq \hocolim X \end{align}$$ using at the end $\colim X \mathop{\times}_{\colim Y} \tilde{Y}_i \approx \tilde{X}_i$ (this is where one needs to open up the Bousfield-Kan constructions $\tilde{X}$ and $\tilde{Y}$, I believe, making use of the isomorphism $X_i \approx Y_i \times_{\colim Y} \colim X$ from above).

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