Categorical systems theory: control and emergence
This will be an informal session with plenty of discussion time, investigating a couple of concepts that arise from the category-theoretic treatment of systems.
(1) Control
How inputs to a system regulate its behaviour. Starting points: (a) In response to the ‘active inference’ program, some recent articles (e.g., https://arxiv.org/abs/2406.07577 and https://arxiv.org/abs/2508.06326) have looked to understand autonomous systems as composed of agent and controller subsystems, equipped with dual interfaces; (b) Ordinary Lyapunov functions have been treated category-theoretically (https://arxiv.org/abs/2502.15276), work that should be extendable to variants. Where ISS (input-to-state stability) Lyapunov functions concern stability under any external perturbation, control Lyapunov functions concern stability under a chosen input.
(2) Emergence
Phenomena where the composite behaviour of the parts does not equate to the behaviour of the composite. Starting points: (a) Elie Adam’s thesis, ‘Systems, Generativity and Interactional Effects’ (https://elieadam.com/eadam_PhDThesis.pdf); (b) Puca et al. on ‘Failures of compositionality’ (https://arxiv.org/abs/2307.14461) (c) Erik Hoel on causal emergence (e.g., https://arxiv.org/abs/2202.01854). Two relevant CT constructions appear to be laxness of functors and coarse-graining as epimorphisms, potentially fitting well with a double category-theoretic outlook.