Fabio PASQUALI (Università di Milano) will give a talk on
Relational quotient completion
Abstract:
The importance of completing a structure with quotients is pervasive in mathematics and several constructions have been refined to allow one to work with quotients even though they are not natively available in the setting in which one is reasoning.
Among the mathematical tools that have been adopted to study quotients, Lawvere’s doctrines offer a simple and powerful framework capable to cope with a large variety of situations. A doctrines P is a contravariant functor from a category C with finite products to the category of posets and monotone functions. Doctrines they provide a functorial description of first order theories in the sense that: objects and arrows of C model contexts and terms, products of C model context concatenation and P maps each object to a poset that models formulas on that object ordered by logical entailment.
In this talk we present relational doctrines as a functorial description of the core fragment of the calculus of relations: here one takes as primitive concepts (binary) relations (instead of unary predicates) and as basic operations relational composition, relational identities and the converse of a relation.
Relational doctrines provide a natural setting where to deal with quotients. We then describe a universal construction that adds quotients to any relational doctrine (inspired by Maietti and Rosolini elementary quotient completion) showing that this construction subsumes many known examples and has new ones, such as the category of metric spaces and non-expansive maps. Finally we give a characterization of relational doctrines that arise as a quotient completion as those having quotients and enough projectives, commenting the connection of this characterization with similar ones by Carboni and Vitale on the exact completion of a weakly lex category and on Maietti and Rosolini construction.
This is a joint work with Francesco Dagnino (University of Genoa).