Classifying Toposes for First Order Theories
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By a classifying topos for a first-order theory T, we mean a topos<br />E such that, for any topos F, models of T in F correspond exactly to<br />open geometric morphisms F ! E. We show that not every (infinitary)<br />first-order theory has a classifying topos in this sense, but we<br />characterize those which do by an appropriate `smallness condition',<br />and we show that every Grothendieck topos arises as the classifying<br />topos of such a theory. We also show that every first-order theory<br /> has a conservative extension to one which possesses<br /> a classifying topos, and we obtain a Heyting-valued completeness<br /> theorem for infinitary first-order logic.
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2001 ◽
Vol 11
(1)
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pp. 21-45
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