This question may be a bit vague, (so if suggested, can I make it community wiki), but I was wondering what techniques there exists for computing homotopy colimits in a category of fibrant objects. A category of fibrant objects is great for computing homotopy limits of course, but I'm not sure what, if anything can be said about homotopy colimits. Given a colimiting cocone in a category of fibrant objects $(C,W,F)$, are there any conditions that ensure this is actually a homotopy colimit? (Lets agree that by homotopy colimit, I mean a colimit in the associated $\left(\infty,1\right)$-category). Of course, I know what to say if $(C,W,F)$ is really a Quillen model category, but lets assume that it is not.
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2$\begingroup$ Being equally vague, I'd suggest to look at this question from the point of view of derivators. Categories of fibrant objects (depending on the axioms you take) have an associated right derivator (right/left also depends on the author, but I mean derivator with htpy limits). The existence of htpy colimits for some diagrams is just a property which this derivator may (or not) have. $\endgroup$– Fernando MuroCommented Jul 8, 2014 at 15:15
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$\begingroup$ Do you have any suggested starting point? (E.g. an explanation of how a category of fibrant objects is a derivator). I haven't yet learned the derivator formalism. $\endgroup$– David CarchediCommented Jul 8, 2014 at 15:19
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$\begingroup$ take a look at math.univ-toulouse.fr/~dcisinsk/kthwt.pdf It's very long, but you can skip many technical parts. Maybe you can first take a look at the informal discussion in the nlab: ncatlab.org/nlab/show/derivator $\endgroup$– Fernando MuroCommented Jul 8, 2014 at 15:30
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