Lax functors and coalgebraic weak bisimulation

Abstract

We generalize the work by Sobociński on relational presheaves and their connection with weak (bi)simulation for labelled transistion systems to a coalgebraic setting. We show that the coalgebraic notion of saturation studied in our previous work can be expressed in the language of lax functors in terms of existence of a certain adjunction between categories of lax functors. This observation allows us to generalize the notion of the coalgebraic weak bisimulation to lax functors. We instantiate this definition on two examples of timed systems and show that it coincides with their time-abstract behavioural equivalence. Contents1. Introduction 1 Content and organization of the paper 3 2. Basic notions 3 2.1. Coalgebras 4 2.2. Monads and their Kleisli categories 4 2.3. Coalgebras with internal moves 7 2.4. Order enriched categories 7 2.5. Coalgebras and functional simulations 8 2.6. Coalgebraic (weak) bisimulation and saturation 9 2.7. Relational presheaves 12 3. Lax functors 13 3.1. Examples and their properties 13 3.2. Change-of-base functor and its left adjoint 18 4. Weak bisimulation 24 4.1. Weak bisimulation for lax functors 24 4.2. Weak bisimulation on M -flows 28 4.3. Weak bisimulation for coalgebras revisited 30 5. Summary 30 References 31