Examples of categorical transition systems [mde-GAGL]

We give some examples of categorical transition system in the sense of Errington.

• Pointed graphs as the category of shapes, with \(\kappa = U;F\) forgetting the point and taking the free category, and \(\mathbb {C} = \Sigma ^{*}\). These are labelled transition systems with an initial state.
• Graphs as the category of shapes, with \(\kappa \) the free category functor and \(\mathbb {C} = \mathsf {Rel}\). These are program flowcharts in the sense of Burstall, assigning "assertions" (or sets satisfying assertions) to each program point and relations corresponding to "commands".
• \(\mathbf {Cat}\) as the category of shapes with \(\kappa \) the twisted arrow construction. A functor \(\kappa J \to \mathbb {C}\) is essentially an oplax \(J \to \mathbf {Span}\), generalizing Burstall's semantics, c.f. Chapter 5 of Errington.