Wednesday, November 27th, 2019, 15:30, Sala riunioni secondo piano.
Davide Trotta (Università degli Studi di Trento)
Doctrines in categorical logic and the existential completion
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 and existential doctrine, which generalize that of hyperdoctrine, and we present the 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 extend the notion of exact completion of an elementary existential doctrine to an arbitrary elementary doctrine.
Wednesday, November 27th, 2019, 16:45, Sala riunioni secondo piano.
Branko Nikolić (Macquarie University, Sydney, Australia)
On Directed Colimits of Hilbert Spaces
Abstract: We will show that the category of Hilbert spaces and contractions has directed colimits, and that tensoring preserves them. We will also discuss (problems with) the generalization to bounded maps.