# A construction with homotopy colimits and homotopy pullbacks for descent.

I have some troubles in trying to give a meaningful interpretation to the following property which is stated in this preprint by professor Rezk (see Definition 6.5) as part of the requirement for a model category $\mathbf{M}$ to have what the author calls descent. Here is the problematic condition:

(P1) Let $X\colon I\longrightarrow \mathbf{M}$ be a functor from a small category $I$ to a model category $\mathbf{M}$. Set $$\bar{X}:=\text{hocolim}X$$ and consider any map $f\colon\bar{Y}\rightarrow\bar{X}$ in $\mathbf{M}$. Form a functor $Y\colon I\longrightarrow\mathbf{M}$ by setting, for $i\in I$, $$Y(i):=X(i)\times^{h}_{\bar{X}}\bar{Y}.$$ Then the evident map $$\text{hocolim Y}\rightarrow \bar{Y}$$ is a weak equivalence.

(Here $(-)\times^{h}_{\bar{X}}(-)$ denotes a homotopy pullback over $\bar{X}$, I guess).

My doubts arise as I do not see a clear way to understand the construction made. Here are some points that are obscure to me:

1. To have a chance to construct something as $Y(i)$ or the "evident" map $\text{hocolim}Y\rightarrow\bar{Y}$, I should be able to get maps $X(i)\rightarrow \bar{X}$ and $Y(i)\rightarrow\bar{Y}$. The latter family of maps should then induce the alleged arrow $\text{hocolim}Y\rightarrow\bar{Y}$, or at least this is what I believe. The point now is: are there some models for the homotopy colimits and the homotopy pullbacks (for which they form functors from the appropriate diagram categories) that allow me to get maps $X(i)\rightarrow \text{hocolim}X$ and $Y(i)\rightarrow\bar{Y}$ (as for ordinary colimits and pullbacks)? Can these maps be built so as to form a (co)cone (over $\text{hocolim}X$ or over $\bar{Y}$) either in a strict sense or in some homotopically relaxed version (to be defined), so that I can use, for example, the maps $Y(i)\rightarrow\bar{Y}$ to get $\text{hocolim}Y\rightarrow\bar{Y}$?

2. Is the evidence of the map $\text{hocolim}Y\rightarrow\bar{Y}$ really so evident?

3. Are there some canonical properties of the homotopy colimits or of the homotopy pullbacks which allow me to make the construction done by Rezk (and to answer the questions in 1.) in a way which is independent of the chosen model to realize those homotopy colimits and pullbacks?

Note: Rezk considers arbitrary model categories when formulating the property (P1) above, but I would be very happy to get an answer to my questions (which ultimately boil down to the final question: "How should I interpret property (P1)?") in the case where $\mathbf{M}$ is (at most) a combinatorial model category (but not simplicial, if possible).

There is a somehow related question on MO which suggests, in the case where $\mathbf{M}$ is a category of spaces (say, simplicial sets), to feel free to cofibrantly and fibrantly replace everything (objects, arrows, diagrams) as long as this may be needed to give a meaning to constructions like the one in (P1) above. I have then tried to see if I could work out something with this underlying philosophy.

Here are my trials: in what follows, I assume $\mathbf{M}$ to be a cofibrantly generated category and keep the same notations as in (P1) above.

Let $Q$ be the functorial cofibrant replacement in the projective model structure on $\mathbf{M}^{I}$ and let $R$ be the fibrant replacement functor for the Reedy model structure on the category of cospans in $\mathbf{M}$. Given a cospan $$A\rightarrow B\leftarrow C$$ in $\mathbf{M}$, I will adopt a little abuse of notation and denote the image of this cospan under $R$ by $$RA\rightarrow RB\leftarrow RC.$$

Now, a model for the homotopy colimit of $X$ is given by $\bar{X}=\text{colim}\ QX$. In this case, I thus get a cocone $$(QX)(i)\rightarrow \text{hocolim} X$$ and, for each $i\in I$, I can consider the cospan in $\mathbf{M}$ $$(QX)(i)\rightarrow \text{hocolim} X \leftarrow \bar{Y}$$ A model for the homotopy pullback of this cospan is given by doing the ordinary pullback of $$R((QX)(i))\rightarrow R(\bar{X})\leftarrow R(\bar{Y})$$ Thus, my (corrected) functor $Y$ is given by $$Y(i):=R((QX)(i))\times_{R(\bar{X})}R(\bar{Y})$$ and I have an evident natural transformation $$Y\Rightarrow c(R(\bar{Y})),$$ where $c$ is the constant functor from $I$ at $R(\bar{Y})$. Thus, I get a map $$\text{hocolim}(Y)=\text{colim}(QY)\rightarrow \text{hocolim}(c(R(\bar{Y})))=\text{colim}(Q(c(R(\bar{Y})))),$$ hence also a map $$t\colon\text{hocolim}(Y)\rightarrow R(\bar{Y})$$ because I have a natural transformation $Q(c(R(\bar{Y})))\Rightarrow c(R(\bar{Y}))$ and the colimit of the RHS has a canonical map to $R(\bar{Y})$. I may then require this $t$ to be a weak equivalence.

Question: Does this make any kind of sense?

I would really appreciate any kind of partial answer to any of my questions above or even some general suggestions about how to interpret these kinds of constructions with homotopy colimits, pullbacks and so on.