# Codensity Games for Bisimilarity

## TL;DR

This paper introduces a categorical framework for bisimilarity games that unifies and extends game characterizations of bisimilarity and its quantitative variants across various system types.

## Contribution

It provides a formal fibrational and coalgebraic framework that derives codensity bisimilarity games, covering known and new bisimilarity notions.

## Key findings

- Unified categorical framework for bisimilarity games
- Derivation of codensity bisimilarity games
- Applicability to both classical and quantitative systems

## Abstract

Bisimilarity as an equivalence notion of systems has been central to process theory. Due to the recent rise of interest in quantitative systems (probabilistic, weighted, hybrid, etc.), bisimilarity has been extended in various ways: notably, bisimulation metric between probabilistic systems. An important feature of bisimilarity is its game-theoretic characterization, where Spoiler and Duplicator play against each other; extension of bisimilarity games to quantitative settings has been actively pursued too. In this paper, we present a general framework that uniformly describes game characterizations of bisimilarity-like notions. Our framework is formalized categorically using fibrations and coalgebras. In particular, our characterization of bisimilarity in terms of fibrational predicate transformers allows us to derive codensity bisimilarity games: a general categorical game characterization of bisimilarity. Our framework covers known bisimilarity-like notions (such as bisimulation metric) as well as new ones (including what we call bisimulation topology).