Categories for the working modal logician
Giovanni CinĂ
Abstract:
We attempt to build a ladder connecting the heavens of Category Theory to the interests of modal logicians, in particular those concerned with modelling tasks and therefore involved with specific models and languages. Rather than providing a general argument or theory, we set out to collect enough evidence for this fruitful interplay.
The body of work presented in this theses witnesses two possible modes of interaction. The first is the study of hybrid models, namely structures that are on one hand significant from a category-theoretic perspective and on the other hand lend themselves to a treatment with modal languages.
We show how presheaf models can be seen as particular relational structures and develop a hierarchy of modal languages to express their properties.
To argue in favor of the flexibility of this framework we review several applications; we especially dive into the details of a modal logic for social choice functions.
Furthermore, we highlight how in this setting some of the traditional issues of Modal Logic, e.g. completeness, expressivity and decidability, receive an original twist and can be resolved with alternative solutions.
A second mode, more heuristic in nature, consists of regarding a given class of models as a category. The benefit of this stance is the baggage of questions that come with it. The right notion of morphism for these models, its closure under composition, the functoriality of some uniform constructions, these are some of the basic issues that get raised in this context. In the second half of the thesis we explicate how they can shape research in Modal Logic and how they are intertwined with existing problems.
We believe that the examples we treated and the techniques we introduced are not isolated success stories, but rather an indication that the interaction between Category Theory and Modal Logic can be further developed and give rise to a broad scale of further applications.