Homotopy (co)limit (co)cones

Let $\mathscr{M}$ be a model category and let $\mathscr{I}$ be a small category. Consider any homotopy colimit functor $\text{hcolim}_{\mathscr{M}}^{\mathscr{I}}\colon\mathscr{M}^{\mathscr{I}}\longrightarrow \mathscr{M}$ of shape $\mathscr{I}$ on $\mathscr{M}$. Here I adopt the notion of homotopy colimit considered in Homotopy Limit Functors on Model Categories and Homotopical Categories where a homotopy colimit is defined as a left approximation of the colimit functor.

Given a functor $X\colon\mathscr{I}\longrightarrow \mathscr{M}$, let me call a cocone $X\rightarrow P$ from $X$ (that is, a natural transformation $X\rightarrow cP$, where $P\in\mathscr{M}$ and $c$ is the constant functor $\mathscr{M}\longrightarrow \mathscr{M}^{\mathscr{I}}$) a homotopy colimit cocone if the canonical composite map $$\text{hcolim}_{\mathscr{M}}^{\mathscr{I}}X\rightarrow \text{colim}_{\mathscr{M}}^{\mathscr{I}}X\rightarrow P$$ is a weak equivalence. Note that, given another homotopy colimit functor $\textbf{hcolim}_{\mathscr{M}}^{\mathscr{I}}$ of shape $\mathscr{I}$ over $\mathscr{M}$, the map $\text{hcolim}_{\mathscr{M}}^{\mathscr{I}}X\rightarrow P$ is a weak equivalence if and only if $\textbf{hcolim}_{\mathscr{M}}^{\mathscr{I}}X\rightarrow P$ is a weak equivalence because $\text{hcolim}_{\mathscr{M}}^{\mathscr{I}}$ and $\textbf{hcolim}_{\mathscr{M}}^{\mathscr{I}}$ are naturally weakly equivalent as functors over the colimit functor $\text{colim}_{\mathscr{M}}^{\mathscr{I}}$.

Let me now say that a functor $F\colon \mathscr{M}\longrightarrow \mathscr{N}$ between model categories preserves homotopy colimit cocones if, for every $X\in\mathscr{M}^{\mathscr{I}}$ (for a small category $\mathscr{I}$) and every homotopy colimit cocone $X\rightarrow P$ from $X$, $F\circ X\rightarrow FP$ is a homotopy colimit cocone from $F\circ X$.

Of course, we can dually define the concepts of homotopy limit cones and of preserving homotopy limit cones.

I would like to know if these notions are somehow well understood and/or useful. I am in particular interested in the following questions which I can not answer:

1) Let $F\colon \mathscr{M}\longrightarrow \mathscr{N}$ be a left Quillen functor and let $Q\colon\mathscr{M}\longrightarrow \mathscr{M}$ be a cofibrant replacement functor on $\mathscr{M}$. Does $F\circ Q$ preserve homotopy colimit cocones (and dually for a right Quillen functor)?

2) If $F\colon \mathscr{M}\longrightarrow \mathscr{N}$ is a left Quillen functor which is part of a Quillen equivalence and $Q$ is a cofibrant replacement functor on $\mathscr{M}$, does $F\circ Q$ preserve homotopy limit cones (and dually for a right Quillen functor)?

• I don't think your definition of homotopy colimit cocone is correct. For instance, consider any diagram in $\mathbf{sSet}$ whose homotopy colimit is non-contractible but whose strict colimit is a point. – Zhen Lin May 4 '15 at 20:15
• Dear @ZhenLin, I guess I can see what you would like to suggest. However, I do not fully understand what you mean when you say that the definition is not "correct". Maybe I should have said it before, but I used the terminology "homotopy colimit cocone" just because I had to find a way to call a cocone having that property. However, I am not claiming that my definition should coincide with another one that perhaps is already conventional (I do not know if this is the case).[...] – Marco Vergura May 4 '15 at 20:40
• @ZhenLin I don't think he's asserting that homotopy colimit cocones exist for all diagrams. – Tyler Lawson May 4 '15 at 20:44
• If you add the hypothesis that all objects in $\mathcal{M}$ are cofibrant and take $Q$ to be the identity endofunctor, then what you want is true. The problem with your condition is that it involves an object that is not necessarily cofibrant, so it is impossible to say what the derived functor does to it. – Zhen Lin May 4 '15 at 21:59
• You're simply talking about those homotopy colimits such that the canonical map to the honest colimit admits a section up to homotopy. I don't think one can answer your questions unless you impose conditions on the preservation of colimits by Q. – Fernando Muro May 5 '15 at 1:15