Classifying Toposes for First Order Theories
Carsten Butz July 1997 |

## Abstract: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 (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
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