CLASSIFYING TOPOSES FOR FIRST-ORDER THEORIES

By a classifying topos for a first-order theory T, we mean a topos E s uch that, for any topos F, models of T in F correspond exactly to open geometric morphisms F --> E. We show that not every (infinitary) firs t-order theory has a classifying topos in this sense, but we character ize those which do by an appropriate 'smallness condition', and we sho w that every Grothendieck topos arises as the classifying topos of suc h a theory. We also show that every first-order theory has a conservat ive extension to one which possesses a classifying topos, and we obtai n a Heyting-valued completeness theorem for infinitary first-order log ic.