Two Homotopy Colimit Questions

Solution 1:

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.)

Solution 2:

For (1), you might try figuring out whether the natural map $hocolim_i \rightarrow colim_i$ is an equivalence. For example, you may be in luck and your diagram may be cofibrant in the projective model structure on diagrams.

Alternatively, a homotopy pushout can be constructed using a double mapping cylinder, and it seems not too difficult to verify directly that commutes with ordinary colimits. You might want to be careful when making a general statement about this, because sometimes when people say hocolim they're thinking of different definitions that are only weakly equivalent; if this is the case then the homotopy type of $colim_i hocolim_j F(i,j)$ is not even well-defined. I believe that with the usual bar construction definition of hocolim (which gives a double mapping cylinder in the case you want), at least on a finite diagram, these operations actually do commute. (Hopefully I'm not messing up some subtle point-set topology in saying this.)

Solution 3:

For (2), the fact that left Quillen functors are compatible with homotopy colimits is shown in great generality in:

Dwyer, Hirschhorn, Kan, Smith: Homotopy Limit Functors on Model Categories and Homotopical Categories

See statement 19.2.