ON COMPLETENESS AND COCOMPLETENESS IN AND AROUND SMALL CATEGORIES

The simple connection of completeness and cocompleteness of lattices g rows in categories into the Adjoint Functor Theorem, The connection of completeness and cocompleteness of Boolean algebras - even simpler - is similarly related to Pare's Theorem for toposes. We explain these r elations, and then study the fibrational versions of both these theore ms - for small complete categories. They can be interpreted as definab ility results in logic with proofs-as-constructions, and transferred t o type theory.