  We give an analysis and generalizations of some long-established constructive
completeness results in terms of categorical logic and pre-sheaf and sheaf
semantics. The purpose is in no small part conceptual and organizational: from
a few basic ingredients arises a more unified picture connecting constructive
completeness with respect to Tarski semantics, to the extent that it is
available, with various completeness theorems in terms of presheaf and sheaf
semantics (and thus with Kripke and Beth semantics). From this picture are
obtained both ("reverse mathematical") equivalence results and new constructive
completeness theorems; in particular, the basic set-up is flexible enough to
obtain strong constructive completeness results for languages of arbitrary size
and languages for which equality between the elements of the signature is not
decidable.
