# Reference request: colimits of locally presentable categories

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).

• 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. – Philippe Gaucher Dec 18 '13 at 22:33
• Philippe: "cocontinuous" means "preserves colimits". I believe "presentable" is hipster-speak for "locally presentable". – Tom Leinster Dec 18 '13 at 23:50
• 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. 5.5.3.4 and Thm 5.5.3.18. (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) – Dylan Wilson Dec 19 '13 at 0:50
• @TomLeinster: Is "hipster" hipster-speak for "young'uns"? – Theo Johnson-Freyd Dec 19 '13 at 17:55
• 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!) – Tom Leinster Dec 20 '13 at 0:39