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I'm trying to learn about and compute homotopy (co)limits. Specifically, if $\mathcal{C}$ is some Grothendieck site and $\mathcal{P}$ the simplicial model category of simplicial presheaves (say with the Cech model structure)structure in which weak equivalences and cofibrations are defined level-wise), then I would like to understand how to compute the homotopy colimit of the simplicial diagram determined by a covering ${ U_{i} \rightarrow X }$ in $\mathcal{C}$.

In the answer to my previous question, David Carchedi gave a conceptual/$\infty$-categorical explanation of why this homotopy colimit gets identified with the sieve determined by the covering. It would also be nice to see an explicit, model-categorical computation of this homotopy colimit. But I'm almost overwhelmed by the general theory of homotopy colimits in model categories and don't see how to apply the general theory to this example. If I understand correctly, I could use the projective model structure on diagram categories, a cofibrant replacement of the Cech diagram, and then compute the colimit of that. But I don't know what a cofibrant replacement of the Cech diagram looks like, and besides, there is maybe a better way to do this, perhaps using that we are working in a nice simplicial model category, not just any old model category.

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# Computing homotopy (co)limits in a nice simplicial model category?

I'm trying to learn about and compute homotopy (co)limits. Specifically, if $\mathcal{C}$ is some Grothendieck site and $\mathcal{P}$ the simplicial model category of simplicial presheaves (say with the Cech model structure), then I would like to understand how to compute the homotopy colimit of the simplicial diagram determined by a covering ${ U_{i} \rightarrow X }$ in $\mathcal{C}$.

In the answer to my previous question, David Carchedi gave a conceptual/$\infty$-categorical explanation of why this homotopy colimit gets identified with the sieve determined by the covering. It would also be nice to see an explicit, model-categorical computation of this homotopy colimit. But I'm almost overwhelmed by the general theory of homotopy colimits in model categories and don't see how to apply the general theory to this example. If I understand correctly, I could use the projective model structure on diagram categories, a cofibrant replacement of the Cech diagram, and then compute the colimit of that. But I don't know what a cofibrant replacement of the Cech diagram looks like, and besides, there is maybe a better way to do this, perhaps using that we are working in a nice simplicial model category, not just any old model category.