Doctrines, modalities and comonads

Francesco Dagnino,Giuseppe Rosolini

doi:10.1017/s0960129521000207

Francesco Dagnino, Giuseppe Rosolini

Open Access

https://doi.org/10.1017/s0960129521000207

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Abstract

AbstractDoctrines are categorical structures very apt to study logics of different nature within a unified environment: the 2-categoryDtnof doctrines. Modal interior operators are characterised as particular adjoints in the 2-categoryDtn. We show that they can be constructed from comonads inDtnas well as from adjunctions in it, and we compare the two constructions. 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.

Highlights

The approach to logic proposed by F.W
We show that an adjunction in the 2-category of doctrines gives rise to a doctrine with a modal operator
Taking a slightly different perspective, we show that a comonad in the 2-category of doctrines determines a doctrine with a modal operator, this time on the category of coalgebras for the comonad

Summary

Introduction

The approach to logic proposed by F.W. Lawvere via hyperdoctrines has proved very fruitful as it provides an extremely suitable environment where to analyse both syntacic aspects of logic and semantic aspects as well as compare one with the other, see Lawvere (1969, 1970). Their meaning is less clear in a typed logical formalism In this setting, there are various semantics which are interrelated, and we show that many of these are instances of the general situation of an adjunction between two homomorphisms of doctrines. An adjunction between doctrines is very much like an adjunction between categories: roughly, it consists of two doctrines P: C op → Pos and Q: Dop → Pos and two homomorphisms of doctrines connecting them, which should be thought of as an interpretation of P in Q (the left adjoint) and an interpretation of Q in P (the right adjoint) Such a situation can be summarised by a modal logic which uses the logic Q to describe properties of types in C (the base category of P). In Appendix A we sketch an example on how to use our construction to obtain models of the bang modality of linear logic

Interior Operators and Doctrines

Interior Modalities from Adjunctions

Interior Modalities from Comonads

The Global Picture