Consider the 2-category of locally presentable categories, cocontinuous functors, and natural transformations. I believe that this 2-category is 2-cocomplete in the sense of containing all small 2-colimits. One proof is outlined in Mike Shulman's answer to Martin Brandenburg's question 2-colimits in the category of cocomplete categories from a couple years ago. In my paper with Alex Chirvasitu (link), we claim (with a sketch of a proof) that the following works: (i) every cocontinuous functor between presentable categories has a right adjoint, which is continuous and commutes with sufficiently-filtered colimits, and every functor of this type has a left adjoint; (ii) take the diagram whose colimit you want to compute, and consider the corresponding diagram in the category of locally presentable categories, right adjoints, and natural transformations; it suffices to compute the limit of that diagram; (iii) compute the limit of that diagram in the 2-category of all categories, and check that the limit is presentable, and that the functors involved in the limit are continuous and commute with sufficiently filtered colimits.

My question is simply a reference request: is there a paper in the published literature that provides a careful proof that the 2-category of locally presentable categories, cocontinuous functors, and natural transformations is 2-cocomplete? I'd rather cite this fact in my current project than reproduce the above argument, and I don't want to cite a paper (even my own) that only provides a "sketch of proof", if the fact is one I plan to rely on later.

The claim is not in the standard reference by Adamek and Rosicky. The closest there is the fact that the 2-category of accessible categories and functors that commute with sufficiently filtered colimits is 2-complete (and in fact 2-limits can be computed in Cat).

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    $\begingroup$ What is a presentable category ? And what is a cocontinuous functor ? I don't understand (i). If you mean a colimit-preserving functor between locally presentable categories, indeed it has a right adjoint by the dual of the Special Adjoint Functor theorem (take the opposite categories and apply SAFT). And an accessible limit-preserving functor between locally presentable categories is always a right adjoint indeed. This is explained in the book you mention. $\endgroup$ Dec 18, 2013 at 22:33
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    $\begingroup$ Philippe: "cocontinuous" means "preserves colimits". I believe "presentable" is hipster-speak for "locally presentable". $\endgroup$ Dec 18, 2013 at 23:50
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    $\begingroup$ Also, as far as I can tell this is also not proven in the "Handbook of Categorical Algebra" which is the other big reference for such things. It is proven in Higher Topos Theory but that's overkill (I imagine if you wanted a proof that worked only in the ordinary categorical case you could set \infty = 1 throughout). I'm speaking of HTT Cor. and Thm (Note that homotopy limits in the infty-category of infty-categories agree with 2-limits in the 2-category of categories when everything in sight is an ordinary category) $\endgroup$ Dec 19, 2013 at 0:50
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    $\begingroup$ @TomLeinster: Is "hipster" hipster-speak for "young'uns"? $\endgroup$ Dec 19, 2013 at 17:55
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    $\begingroup$ Hi Theo. Here's an argument for retaining "locally". Let's stick to finite presentability. As you know, there's a notion of what it means for an object of a category to be finitely presentable. In particular, you can apply this to objects of CAT. So, this gives a notion of finite presentability for categories. But this is not at all the same as local finite presentability of a category. (Ironically, it's exactly when you're thinking higher-categorically that the terminological distinction becomes important!) $\endgroup$ Dec 20, 2013 at 0:39

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I'm a bit late to the party, but I believe there is a canonical reference for this fact: Greg Bird's 1984 thesis Limits in 2-Categories of Locally Presentable Categories. Although apparently unpublished, Google Scholar lists 20 citations to it. According to this post by Steve Lack to the categories mailing list, Bird proves that the 2-category of locally presentable categories, left adjoint functors, and all natural transformations has all flexible 2-limits and all flexible 2-colimits. This includes all bi(co)limits.

I don't know the history, but I gather that Bird's thesis work was in fact closely related to the very development of the notion of a flexible 2-limit: these are the strict 2-limits whose weights are such that they don't don't actually demand anything too strict, the "homotopically meaningful" strict 2-limits. Lack later showed that in fact they are the cofibrant objects in an appropriate 2-model structure on the 2-category of weights.

Unfortunately, I haven't been able to find a copy of Bird's thesis online. Update: Ross Street has now made it available again (link); see comment below.

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    $\begingroup$ I contacted Ross, and he has kindly agreed to put a scanned copy of Bird's thesis back up again, at least for a while. (And presumably permanently, if Bird can be reached and approves; he is no longer in mathematics.) $\endgroup$
    – Todd Trimble
    Jul 6, 2016 at 0:20
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    $\begingroup$ Thanks, Todd! I was starting to reach out to folks in Australia (who were very helpful!) but you beat me to the source! Thanks to Ross Street. And of course, to Greg Bird. $\endgroup$
    – Tim Campion
    Jul 6, 2016 at 2:53

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