To appear in Compositio Math. C. R. Acad. Representing topoi by topological groupoids. Carsten Butz Logical and Cohomological Aspects of the Space of Points of a Topos. Let T be an infinitary geometric theory which has enough models in Sets to satisfy a completeness theorem.
M interprets function types of first order definable types by exponentials of sheaves.
This topological completeness theorem extends a well known theorem of Makkai and Reyes. Since many interesting cohomology theories arise as the cohomology of some topos with enough points, this result reduces in that cases the cohomology to ordinary sheaf cohomology of topological spaces. But for some other cases this seems to be new.