I have two questions about homotopy colimits:
- What can we say about $\operatorname{hocolim}_j\operatorname{colim}_i F(i,j)$? Iterated homotopy colimits commute, but what can we say when the inner one is a regular colimit? In the case I'm looking at, the homotopy colimit is a homotopy pushout and the colimit is filtered, but I'm quite interested in the general case too. An answer in either the general model theoretic case or the specific case of spaces would be interesting to me.
- Since $\Sigma X$ is the homotopy pushout of $*\leftarrow X\to *$ and homotopy colimits commute, it follows that if $F\colon\mathbf I\to\mathbf{Top}$, then $$\operatorname{hocolim}_i \Sigma F(i)=\Sigma(\operatorname{hocolim}_i F(i))$$
However, it seems to me that we should be able to get this result using the fact that $\Sigma$ is a Quillen left adjoint functor. Left adjoints preserve colimits, and it seems reasonable to suspect that Quillen left adjoints would preserve homotopy colimits. Technically we should be talking about the total left derived functor, so my question should really read: If $T\colon\mathbf C\to\mathbf D$ is a left Quillen functor and $F\colon\mathbf I\to\mathbf C$ is a diagram, is it true that:
$$\operatorname{hocolim}_i (\mathbb LT) F(i)=\mathbb LT(\operatorname{hocolim}_i F(i))$$
Best Answer
One way I've thought about this, if the diagram category is a Reedy category: $\mathrm{hocolim}_iF(i)$ is $\mathrm{colim}G(i)$ where $G$ is a cofibrant replacement for $F$. Then $TG$ is also a cofibrant replacement for $(\mathbb{L}T)F$ (as $T$ sends cofibrations to cofibrations and pushouts to pushouts).
So
$$\mathrm{hocolim}_i (\mathbb{L}T)F(i) = \mathrm{colim}_i TG(i) = T \mathrm{colim}_i G(i) = \mathbb{L}T \mathrm{hocolim}_i F(i).$$
(Edited to make everything homotopical.)