## Dirk Hofmann

## A short history of ultrafilter convergence

Over the past few years, our principal research was concerned with the study of topological structures as generalised categories, interpreting the convergence relation between ultrafilters and points of a topological space as the arrows of a category. This research project has its roots in Lawvere's observation that both ordered sets and metric spaces can be viewed as enriched categories, and in Barr's description of topological spaces as relational algebras for the ultrafilter monad.

In this talk we wish to present ultrafilter characterisations of special classes of spaces and continuous maps (the "early history"), and, as much as time allows, how this work led to the development of a general setting presenting spaces as categories.

In this talk we wish to present ultrafilter characterisations of special classes of spaces and continuous maps (the "early history"), and, as much as time allows, how this work led to the development of a general setting presenting spaces as categories.

Manuel António Martins

## Hybridisation of logics

Hybridisation is a systematic process along which the characteristic features of hybrid logic, both at the syntactic and the semantic levels, are developed on top of an arbitrary logic framed as an institution. It also captures the construction of first-order encodings of such hybridised institutions into theories in first-order logic. The method was originally developed to build suitable logics for the specification of reconfigurable software systems on top of whatever logic is used to describe local requirements of each system’s configuration. Actually, it provides a fresh example of yet another development in combining logics driven by a problem from Computer Science. We will present an overview of the hybridisation method, as well as its motivation and some applications.

This is a joint work with Luís Barbosa and Alexandre Madeira.

This is a joint work with Luís Barbosa and Alexandre Madeira.