Classifying Toposes for First Order Theories

Abstract

By a classifying topos for a first-order theory T, we mean a topos
E such that, for any topos F, models of T in F correspond exactly to
open geometric morphisms F ! E. We show that not every (infinitary)
first-order theory has a classifying topos in this sense, but we
characterize those which do by an appropriate `smallness condition',
and we show that every Grothendieck topos arises as the classifying
topos of such a theory. We also show that every first-order theory
has a conservative extension to one which possesses
a classifying topos, and we obtain a Heyting-valued completeness
theorem for infinitary first-order logic.