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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.

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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?).