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 model 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.
Best Answer
In practice, one tends to "compute" arbitrary homotopy colimits as bar constructions, especially when you have a simplicial model category.
If $X:J\to P$ is a simplicially enriched functor, where $J$ is small, then you get a "bar construction" $B=B(*,J,X)$. This is a simplicial object in $P$, with $$B_0 = \coprod_{j_0\in \mathrm{ob}J} X(j_0),$$ $$B_1=\coprod_{j_0,j_1\in \mathrm{ob}J} X(j_0)\times J(j_0,j_1),$$ $$B_2=\coprod_{j_0,j_1,j_2\in \mathrm{ob}J} X(j_0)\times J(j_0,j_1)\times J(j_1,j_2),$$ etc.
(Here "$\times$" really means the simplicial "$\otimes$"; if $P$ is simplicial sets, then it really is $\times$.) If $X$ is suitably cofibrant, then the realization $|B|$ of $B$ will be the homotopy colimit of $X$.
This bar construction I described above is really a special case of "use the projective model structure"; you can use a bar construction to an explicit construction of a projective cofibrant resolution of $X$ (typically under some hypothesis on $X$, such as that each $X(j)$ is cofibrant in $P$). In fact, $$|B(*,J,X)| = \mathrm{colim}_J |j\mapsto B(J(-,j),J,X)|,$$ and there is a weak equivalence $|B(J,J,X)|\to X$, which is a true projective cofibrant approximation given some mild hypothesis on $X$.
The standard references are oldies but goodies: Segal's paper "Classifying spaces and spectral sequences," IHES 1968, and the "yellow monster": Bousfield & Kan, "Homotopy Limits, Completions, and Localizations," LNM 304.
Added:
When $J=\Delta^{\mathrm{op}}$, you can say something easier: the homotopy colimit of $X: J\to P$ is computed by the realization $|X| \in P$ (again, up to the cofibrancy of the objects $X(j)$). I don't know an explicit reference offhand, though everybody uses this fact; it may be in the two that I cited.