Classifying Toposes for First Order Theories

Carsten Butz
Peter T. Johnstone

July 1997


By a classifying topos for a first-order theory tex2html_wrap_inline22 , we mean a topos tex2html_wrap_inline24 such that, for any topos tex2html_wrap_inline26 , models of tex2html_wrap_inline22 in tex2html_wrap_inline26 correspond exactly to open geometric morphisms tex2html_wrap_inline32 . 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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