Bisimulation maps in presheaf categories

TL;DR

This paper introduces a general framework for bisimulation maps within presheaf categories, providing an abstract and versatile approach to capturing various behavioral equivalences in dynamical systems.

Contribution

It defines bisimulation maps in presheaf categories and characterizes standard, fair, and branching bisimulations within this abstract setting.

Findings

Unified framework for different bisimulations

Characterizations for standard, fair, and branching bisimulations

Applicability to various dynamical systems behaviors

Abstract

The category of presheaves on a (small) category is a suitable semantic universe to study behaviour of various dynamical systems. In particular, presheaves can be used to record the executions of a system and their morphisms correspond to simulation maps for various kinds of state-based systems. In this paper, we introduce a notion of bisimulation maps between presheaves (or executions) to capture well known behavioural equivalences in an abstract way. We demonstrate the versatility of this framework by working out the characterisations for standard bisimulation, $\forall$-fair bisimulation, and branching bisimulation.