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Filtered Colimit Preserving Functors on Models of a Regular Theory

This note recalls the representation of regular theories T in terms of set-valued functors on models given by Makkai(1990), and explicitly states the representation theorem for the classifying topos Set[T] in terms of filtered colimit preserving functors which can be extrapolated from the results of that paper. That representation of Set[T] is then compared with topological representations in the style of Butz and Moerdijk(1998) by showing that for a certain natural topology on the space of models, preserving filtered colimits is the same thing as being `continuous' in the sense of being an equivariant sheaf. By using a slight variation of the topology originally presented in op. cit., we obtain from this comparison a representation of Set[T] in terms of a topological category of models and homomorphisms, where the restricted topological groupoid of models and isomorphisms classifies a different (non-regular) theory.