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I have two questions about homotopy colimits:

  1. 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.
  2. 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: $$\operatorname{hocolim}_i (\mathbb LT) F(i)=\mathbb LT(\operatorname{hocolim}_i F(i))$$
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2 Answers

up vote 5 down vote accepted

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.)

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Thank you. This was my first thought, but I wasn't sure if TG was the cofibrant replacement for $TF$. I'll have to think about why this happens. –  SL2 Apr 30 '12 at 17:27
    
You need to check two things: 1) That $TG$ is a cofibrant diagram, and 2) That $TG$ is weakly equivalent to $\mathbb{L}TF$. For 1), use that $G$ is a cofibrant diagram and that $T$ preserves colimits, cofibrations and trivial cofibrations. For 2) the $\mathbb{L}$ is necessary since $T$ is not necessarily homotopical. But $\mathbb{L}TF = \mathbb{L}TG = TG$, since $G$ has cofibrant image. –  Thomas Belulovich Apr 30 '12 at 17:36
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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.)

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Hmmm, that's a good point that it isn't even well-defined. So I guess the answer to the first question is 'not much' unless the diagram is cofibrant. –  SL2 Apr 30 '12 at 19:11
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