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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