Defining a notion of canonical extension for coherent categories.
In the 1950s Jonsson and Tarski introduced the notion of canonical extension of a Boolean algebra with operators. Thereafter their ideas have been developed further, which has led to a smooth theory of canonical extensions applicable in a broad setting such as distributive lattices and even partially ordered sets. The theory of canonical extensions has been a powerful tool in the study of propositional logics. My aim is to generalize this theory to the categorical setting, so that it can also be applied in the study of predicate logics.
In this talk I will define the notion of canonical extension of a partially ordered set and give an example of an application of canonical extensions in the study of propostional logic. Thereafter I will explain the basic ideas we have at the moment about extending this theory to the categorical setting.