Given a simplicial combinatorial model category $\mathcal{M}$ and a simplicial diagram $F\colon \Delta^{\mathrm{op}} \rightarrow \mathcal{M}$, is there a nice (i.e. explicitely computable) way of expressing its homotopy colimit? For instance, if one could give a cofibrant replacement $\widetilde{F}$ of $F$ (with respect to the projective model structure) which is quite explicit at least in dimension 0 and 1, by which I mean the value on the restriction $\Delta^{\mathrm{op}}_{\leq 1}$, then the coequalizer of $\widetilde{F}_1\stackrel{\longrightarrow}{\longrightarrow}\widetilde{F}_0$ would do the job.
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5$\begingroup$ If the diagram is Reedy cofibrant then you can just take the geometric realization. (It is much easier to be Reedy cofibrant than it is to be projectively cofibrant, so this is something you’re likely to actually be able to “compute”). $\endgroup$ – Dylan Wilson May 29 at 11:16