Dear all, I wonder how the following 2-categories are related? 1. The 2-category of locally presentable categories and left adjoints. 2. The full sub-2-category of the 2-category of cocomplete categories and cocontinuous functors, consisting of those categories which are small bicolimits (in that 2-category) of diagrams of presheaf categories on small categories. 3. Like (2), but consisting only of those categories which are codescent objects of diagrams of presheaf categories on small categories. Since every locally presentable category is a small-orthogonality class in a presheaf category, I think it follows that it is a coinverter (in the 2-category of cocomplete categories) of a transformation between two presheaf categories. Thus (1) is a subcategory of (2), and a full subcategory by the adjoint functor theorem. It is of course clear that (3) is a full subcategory of (2), and I think they should be the same, since bicolimits can be constructed from coproducts, copowers, and codescent objects, while small coproducts and copowers of presheaf categories are again presheaf categories. It seems likely to me that (1) and (2) are also the same; has anyone studied this question? I am wondering about this because some people have recently started using "presentable category" as a synonym for "locally presentable category," with (as far as I understand) the intuition that the above description of a locally presentable category as a coinverter is a "presentation" of it -- in contrast with the meaning of "locally presentable category" that it is the *objects* of the category which are presentable (in the sense that homming out of them preserves sufficiently highly filtered colimits). I would find the most intuitive sort of "presentation" for an object of a 2-category to be a codescent object of a diagram of free objects, rather than a coinverter; for instance, that is the sort of presentation that seems to arise in pseudo-monadicity theorems. So I wondered whether locally presentable categories are also the categories that can be "presented" as codescent objects of diagrams of presheaf categories on small categories (the latter being, of course, the free cocomplete categories on small categories), and the step to all small colimits is natural. Regards, Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]