By a classifying topos for a first-order theory , we mean a topos such that, for any topos models of in correspond exactly to open geometric morphisms → . We show that not every 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. (shrink)

Let C be a small Barr-exact category, Reg the category of all regular functors from C to the category of small sets. A form of M. Barr's full embedding theorem states that the evaluation functor e : C →[Reg, Set ] is full and faithful. We prove that the essential image of e consists of the functors that preserve all small products and filtered colimits. The concept of κ-Barr-exact category is introduced, for κ any infinite regular cardinal, and the natural (...) generalization to κ-Barr-exact categories of the above result is proved. The treatment combines methods of model theory and category theory. Some applications to module categories are given. (shrink)