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

Venue: Seminar Room “-1” – Department of Mathematics – Via Sommarive, 14  Povo - Trento
Time: 3.00 p.m.

• Davide Trotta - PhD in Mathematics

Abstract:
The notion of hyperdoctrine was introduced by F.W. Lawvere in a series of seminal papers to synthesize the structural properties of logical systems. His crucial intuition was to consider logical languages and theories as indexed categories and to study their 2-categorical properties. For instance, connectives and quantifiers are determined by adjunctions. In this seminar we introduce the notions of primary, elementary and existential doctrine, which generalize that of hyperdoctrine, and we present the generalized existential completion. In particular we show how we can add left adjoints along certain functors to a primary doctrine obtaining an existential doctrine.
Moreover we show that the 2-monad obtained from this free construction is lax-idempotent, and that the 2-category of existential doctrines is isomorphic to the 2-category of algebras for this 2-monad.
Finally we present another application of the previous completion, which is the elementary completion of a syntactic doctrine, and we conclude explaining how one can combine this categorical approach to logic given by the study of doctrines, with the universal algebraic techniques given by the formal theory of the pseudo-monads and pseudo-distributive laws.

Supervisors: Roberto Zunino - Giuseppe Rosolini