Filtered Colimit Preserving Functors on Models of a Regular Theory

TL;DR

This paper explores the representation of regular theories using filtered colimit preserving functors and compares it with topological representations, providing new insights into classifying toposes and model topologies.

Contribution

It explicitly states the representation theorem for the classifying topos of a regular theory using filtered colimit preserving functors and compares it with topological models.

Findings

Filtered colimit preserving functors characterize the classifying topos Set[T].

A natural topology on models equates filtered colimit preservation with continuity.

A topological category of models classifies a non-regular theory.

Abstract

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.