When does a cosimplicial object compute homotopy colimits?

Question

Suppose I want to compute the homotopy colimit of a diagram of spaces. There is a simple way of getting a simplicial space from this diagram, and a theorem tells me that taking the geometric realisation of this gets me the correct answer.

Now suppose I'm trying to compute homotopy colimits in some other model category - the category of baubles, say. I can still get a simplicial bauble from a diagram of baubles, but there is no longer any such thing as geometric realisation. However, I can define a realisation functor relative to a fixed cosimplicial bauble; in my previous example, I was using the cosimplicial space given by the standard n-simplices, but I could have defined a formally identical geometric realisation functor given any cosimplicial space.

My guess is that if my cosimplicial bauble is nice, I get the true hocolim of baubles by doing this. (For instance, I suspect 'simplicial chains on the standard simplices' suffices for the category of complexes of abelian groups.) My question is: what does nice mean here? Is there an easy-to-read set of sufficient conditions?

Answer 1

Dear Saul,

The answer to your question is the subject of chapters 16-19 of Phil Hirschhorn's book Model Categories and their Localizations.

To write out the answer in the general case would be prohibitively time consuming, but I'll write a little bit out.

Definition 19.1.5 Let $M$ be a framed model category, and let $\mathcal{C}$ be a small category. If $X$ is a $\mathcal{C}$-diagram in $M$, then the homotopy limit $\operatorname{holim} X$ is defined to be the equalizer of the maps

$$\prod_{\alpha\in \mathcal{C}} (\widehat{X}_\alpha)^{\mathfrak{N}(\mathcal{C}\downarrow \alpha)}\rightrightarrows \prod_{\sigma:\alpha\to \alpha'\in \operatorname{Arr}(\mathcal{C})} (\widehat{X}_{\alpha'})^{\mathfrak{N}(\mathcal{C}\downarrow \alpha)}$$

Where $\widehat{X}_\alpha$ is the natural simplicial frame on $X_\alpha$.

In particular, the key concept here is the concept of a framed model category. A framed model category is defined to be a model category equipped with the data of cosimplicial and simplicial framing functors $M\to M^\Delta$ and $M\to M^{\Delta^{op}}$, where these are defined as follows:

A cosimplicial frame on an object $X$ is an object $\widetilde{X}$ equipped with a reedy weak equivalence $\widetilde{X}\to cc_\ast X$ where $cc_\ast X$ is the constant cosimplicial object defined by $X$.

A simplicial frame on an object $X$ is an object $\widehat{X}$ equipped with a Reedy weak equivalence $cs_\ast X\to \widehat{X}$ where $cs_\ast$ is the constant simplicial object defined by $X$.

In a framed model category, we require that we have functorial frames. It is proven in Hirschhorn's book that if the original model category has functorial factorizations, then there exist essentially unique (up to a contractible space of choices) functorial simplicial and cosimplicial frames on $M$ (Theorem 16.6.9 and Theorem 16.6.10).

I'm sure you won't have a hard time finding a copy (isn't Hirschhorn at MIT?).

If your model category is either combinatorial or cofibrantly generated (don't remember which, but I think you only need cofibrant generation. I believe that the defect with non-combinatorial cofibrantly generated model categories is that the diagram categories are not necessarily cofibrantly generated again, while combinatorial model categories actually are stable under exponentiation by small categories.), you can also define the holim and hocolim to be derived functors of the ordinary ones, since lim and colim give a quillen adjunction between the model structure on $M$ and its projective and injective diagram model categories.