Classifying Toposes for First Order Theories

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.