Doctrines, modalities and comonads

TL;DR

This paper explores the categorical structures of doctrines and modal interior operators, demonstrating their construction from comonads and adjunctions within the 2-category Dtn, and analyzing the information loss in these processes.

Contribution

It establishes a unified categorical framework connecting doctrines, modal operators, comonads, and adjunctions, extending foundational work by John Power.

Findings

Modal interior operators are characterized as adjoints in Dtn.

Constructs from comonads and adjunctions are compared.

Quantifies information loss from comonads/adjunctions to modal operators.

Abstract

Doctrines are categorical structures very apt to study logics of different nature within a unified environment: the 2-category Dtn of doctrines. Modal interior operators are characterised as particular adjoints in the 2-category Dtn. We show that they can be constructed from comonads in Dtn as well as from adjunctions in it, and the two constructions compare. Finally we show the amount of information lost in the passage from a comonad, or from an adjunction, to the modal interior operator. The basis for the present work is provided by some seminal work of John Power.