Reference. Twisted systems [errington-1999-twisted-systems]

January 31, 2025
2024
David Lindsay Errington

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Definition. Categorical transition system [mde-EM0L]

January 31, 2025
Matt Earnshaw
[errington-1999-twisted-systems]

Given a category \(\mathsf {Shp}\) of system shapes (e.g. graphs) and a functor \(\kappa : \mathsf {Shp} \to \mathsf {Cat}\) and a category \(\mathbb {C}\) ("labels'' or "control paths") a categorical transition system in the sense of Errington comprises,

A shape \(J \in \mathsf {Shp}\),
a functor \(S : \kappa {J} \to \mathbb {C}\).

The appropriate category of such systems is a lax comma (bi)category \(\kappa \downarrow \mathbb {C}\).

Examples of categorical transition systems [mde-GAGL]

January 31, 2025
Matt Earnshaw
[errington-1999-twisted-systems]

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.