Classifying toposes for first-order theories Original Research Article

By a classifying topos for a first-order theory View the MathML source, we mean a topos ∄ such that, for any topos View the MathML source models of View the MathML source in View the MathML source correspond exactly to open geometric morphisms View the MathML source → View the MathML source. 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.