the vague idea that i have at the moment is that a parameterized or "generalized" model of an algebraico-geometric theory is "flat" (in the sense of the inverse image functor preserving kernels) iff the map of moduli stacks is a "fibration" ... or something like that... and that "everything is homotopy-flat" (in the sense of the inverse image functor preserving homotopy-kernels) is essentially the same idea as "every map is a fibration, up to fibrant replacement" ... ??or something... ???....

??so what about the relationship between enriching the algebraico-geometric doctrine in such a way as to constrain the ag morphisms to be fibrations (or something...) and enriching the geometric doctrine in such a way as to constrain the geometric morphisms to be local homeomorphisms or open continuous morphisms or something?? ... and so forth... ??is there some kock-zoeberlein phenomenon here?? or something??...

??hmm, not clear to me yet how similar flatness of an ag morphism is to "fibrationness" in some sense .... ???...

??so what about "_faithfully_ flat" here?? .... hmmm...