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David Corfield (May 26 2026 at 10:57):I've recently been coming across references to HDAs, [[higher-dimensional automata]] (an nLab page I've just started, and in need of expansion). I'm wondering if there's any ACT treatment of them available.
For one thing, they've been related to Petri nets as means of depicting concurrent firing events:
Since they're higher-dimensional versions of ordinary automata, they might seem to offer an opportunity to bring together the Moore-like approach to systems theory and the Petri net-like.
Since HDAs are (pre)-cubical sets of a kind, some forms of presheaves over some cubical category, I'd like to see if its possible to link them up to Petri nets via a morphism of algebraic patterns. My recent foray into Petri nets meet algebraic patterns, Segal presheaves, wheeled props, etc. had me wondering if something could be done with the idea of a restriction along . E.g., we could pull back (oriented) graphical species to some form of cubical species, etc.
Chu and Haugseng provide a potential candidate cubical pattern (Enriched homotopy-coherent structures, p. 32)
One issue is that people tend to opt for precubical sets to define HDAs, avoiding the degeneracy maps since it's thought that there's a better way to represent stuttering steps.