by Carsten Butz, Peter Johnstone
Annals of Pure and Applied Logic
http://www.pmms.cam.ac.uk/preprint/1997/1997-1.ps
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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. 0
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