The points of canonical extensions of doctrines

Abstract

We analyse the space of points of the canonical extension of a coherent doctrine. We first give a full characterisation of doctrine morphisms that are extensible, and relate it to the existing notion of p-model of a coherent category. Through this characterisation, the extensible morphisms are shown to be exactly those which are ω-saturated in the sense of coherent first-order logic. Next, we answer the question: when does a presheaf of models fully describe the canonical extension? We prove a characterisation theorem via two conditions, which are again natural from the perspective of coherent logic, namely, homogeneity and the realisation of all prime types in a strict sense. The characterisation theorem allows us to deduce a reconstruction result for any coherent theory with the property that all prime types can be realised in a countable, saturated model. For instance, ω-stable coherent theories always have this property. We conclude by explaining how our results can be interpreted topos-theoretically, by relating them to the classifying topos and to the topos of types.