Algebra, Logic and Topology Seminar

Codensity monads provide a universal method to generate complex monads from simple functors. Recently, important monads in logic, denotational semantics, and probabilistic computation (such as ultrafilter monads, the Vietoris monad, and the Giry monad) have been presented as codensity monads, using complex arguments. We simplify these codensity presentations by proposing a unifying categorical approach, which only uses density (of functors) and duality (of categories). To sum up: "Codensity Monads = Density + Duality".

We will discuss a non-pointed version of the notion of torsion theory, in the framework of categories equipped with a posetal monocoreflective subcategory such that the coreflector inverts monomorphisms. We will explore the relationships of such torsion theories with factorization systems and categorical Galois structures. We will describe several examples of such torsion theories, in the duals of elementary toposes, in varieties of universal algebras used as models for non-classical logic, and in coslices of the category of abelian groups. Joint work with Andrea Cappelletti.