Abstract: The context is a "finite limit theory" that is, a small category C with finite limits, and a subcanonical topology J on C generated by finite covering families; a coherent category T ("coherent theory") gives an example. Coste's theorem says, roughly, that every "model of C". i.e a lex functor M:C→Set, can be represented as the global sections of a sheaf whose stalks are "models of (C,J)", i.e, functors N:C→Set, "preserving" the coverings in J (in case of a coherent category T, N:T→Set is a coherent functor). This result of Coste's is mentioned in the paper [MM-A.M. Pitts]: "Some results on locally finitely presentable categories", Trans AMS 1987, but no proof is included; I am not aware of any published account of Coste's theorem. I am going to show, mainly, my own proof that I had found early on, without being aware of Coste's work. The apropos for this talk is the beautiful work on the limit closure of various categories of integral domains by Michael Barr, John Kennison and Bob Raffael, which, in the last-heard installment, included a specific new sheaf-representation result. I may be able to say (although I am not sure yet) some reasonable general things (conjectures only at the moment) about the limit closure of the category of models of a coherent theory.
The substance of the talk appears below, the 6th item in the list of supplemental documents.
I am enclosing five items. The 2nd and the 3rd are slides from talks (Dalhousie October and McGill September) - maybe good to get an idea of what I have been doing. The 1st, 4th and 5th are detailed proofs of some of the things I was saying in the talks. The 4th and 5th are two instalments of one thing, the proof of the statement that follows below. The 1st item is needed for the last two.
The most quotable result is: the category Hom, whose objects are small bicategories, and whose morphisms are the homomorphisms of bicategories is equivalent to the category of regular functors on a small (countable) regular category to Set - in particular, Hom is aleph-one accessible (although not aleph-zero accessible (I believe)), with filtered (i.e., aleph-zero-filtered) colimits.