# Coalgebraic Geometric Logic: Basic Theory

## TL;DR

This paper develops a coalgebraic framework for geometric modal logic, integrating topological models and predicate liftings, and analyzes their equivalences and categorical properties.

## Contribution

It introduces a uniform coalgebraic approach to geometric modal logic, including model construction, derivation systems, and equivalence notions, extending the theory to topological spaces.

## Key findings

- Established soundness and completeness of the logic.
- Provided a method to lift endofunctors to topological spaces.
- Compared modal, behavioural, and bisimulation equivalences.

## Abstract

Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor on some full subcategory of the category of topological spaces and continuous functions. We investigate derivation systems, soundness and completeness for such geometric modal logics, and we specify a method of lifting an endofunctor on Set, accompanied by a collection of predicate liftings, to an endofunctor on the category of topological spaces, again accompanied by a collection of (open) predicate liftings. Furthermore, we compare the notions of modal equivalence, behavioural equivalence and bisimulation on the resulting class of models, and we provide a final object for the corresponding category.