Existential completion and pseudo-distributive laws: an algebraic approach to the completion of doctrines

The main purpose of this thesis is to combine the categorical approach to logic given by the study of doctrines, with the universal algebraic techniques given by the theory of the pseudo-monads and pseudo-distributive laws. Every completions of doctrines is then formalized by a pseudo-monad, and then combinations of these are studied by the analysis of the pseudo-distributive laws. The starting point are the works of Maietti and Rosolini, in which they describe three completions for elementary doctrines: the first which adds full comprehensions, the second comprehensive diagonals, and the third quotients. Then we determine the existential completion of a primary doctrine, and we prove that the 2-monad obtained from it is lax-idempotent, and that the 2-category of existential doctrines is isomorphic to the 2-category of algebras for this 2-monad. We also show that the existential completion of an elementary doctrine is again elementary and we extend the notion of exact completion of an elementary existential doctrine to an arbitrary elementary doctrine. Finally we present the elementary completion for a primary doctrine whose base category has finite limits. In particular we prove that, using a general results about unification for first order languages, we can easily add finite limits to a syntactic category, and then apply the elementary completion for syntactic doctrines. We conclude with a complete description of elementary completion for primary doctrine whose base category is the free product completion of a discrete category, and we show that the 2-monad constructed from the 2-adjunction is lax-idempotent.