Mamuka Jibladze wrote:
A related question: does anybody know any analogs of the Freyd's Adjoint Functor Theorems for functors between in(co)complete categories?
Borceux states the 'More General Adjoint Functor Theorem' in Vol 1, 6.6.1 of his Handbook. This requires only that the codomain of the hoped-for left adjoint is Cauchy-complete (and of course that the known functor has some properties: it is 'absolutely flat' and satisfies some solution set conditions). Here's a representability theorem, presumably related. Let C be a small, Cauchy-complete category and let X: C ---> Set. Then X is representable <=> X respects small limits. The same goes for familial representability and connected limits. Proofs are at http://www.ihes.fr/~leinster/rr.ps Tom