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Is there an explicit proof anywhere in the literature that filtered 2-colimits commute with finite 2-limits (all meant in the weak bicategorical sense) in the 2-category of groupoids? I have only been able to find a handful of papers about filtered 2-colimits, notably Dubuc and Street's A construction of 2-filtered bicolimits of categories, and have not found this result in any of them.

Of course one could try to deduce this from $\infty$-categorical results, but the latter proofs are also often sketchy, and in the case of groupoids it ought to be possible to give a concrete constructive proof.

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    $\begingroup$ In case it's useful, Theorem 7.24 of Canevali's thesis 2-filtered bicolimits and finite weighted bilimits commute in Cat proves the result for categories rather than groupoids. Perhaps the reflectivity of Grpd in Cat is enough to transfer this to groupoids? $\endgroup$
    – varkor
    Commented Apr 27, 2021 at 0:00
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    $\begingroup$ @varkor Thanks, that's terrific! And the citation to Dupont's Interchange of filtered 2-colimits and finite 2-limits might be even better. I expect reflectivity isn't even needed if the filtered domains are 1-categories, since groupoids are closed in Cat under all limits and under colimits over 1-categories. If you post your comment as an answer I would probably accept it. $\endgroup$
    – Mike Shulman
    Commented Apr 27, 2021 at 2:05
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    $\begingroup$ @MikeShulman Lurie reduces to the assertion that homotopy cartesian squares of Kan complexes are stable under colimits indexed by directed partially ordered sets, which he refers to as a "classical fact". It's true that such facts about simplicial homotopy theory are regularly used without citation in HTT, but I think they really are classical. For instance here, you just have to observe that since you've got all the maps you need, you can detect homotopy cartesianness by looking at homotopy groups, and that homotopy groups of Kan complexes commute with filtered colimits. $\endgroup$
    – Tim Campion
    Commented Apr 27, 2021 at 10:41
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    $\begingroup$ The classical result is for strictly commutative diagrams, however, so somewhere one is using the fact that diagrams can be strictified… $\endgroup$
    – Zhen Lin
    Commented Apr 27, 2021 at 10:52
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    $\begingroup$ @ZhenLin That seems valid. I think the simplest patch is to appeal to Prop 4.2.4.4 (which was used earlier in the proof in a similar way) to see that any functor of $\infty$-categories $\square \times A \to Spaces$ may be represented by a functor of 1-categories $\square \times A \to Kan$, where $\square = [1] \times [1]$ is the walking commutative square and $Kan$ is the category of Kan complexes (and $A$ is the directed poset as in the proof). $\endgroup$
    – Tim Campion
    Commented Apr 27, 2021 at 22:36

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Two relevant papers are:

The former proves that finite conical pseudolimits and filtered pseudolimits commute in $\mathbf{Cat}$; whereas the latter proves the analagous result for finite weighted bilimits and filtered bicolimits.

It may be that these are enough to recover commutivity in $\mathbf{Grpd}$ for the cases in which you are interested.

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