Myers' categorical system theory is a double categorical yoga for describing the compositional structure of open dynamical systems. It unifies and builds on previous work on operadic notions of system theory, and provides a strong conceptual scaffolding for behavioral system theory. However, some of the most interesting systems out have a richer compositional structure than that of dynamical systems. These are cybernetic systems, or in other words, interactive control systems. Notable and motivating examples are strategic games and machine learning models. In this talk I'm going to introduce the tools and language of categorical system theory and outline how categorical cybernetics theory might look like. At the end, we will briefly venture into the triple dimension.

Categorical system theory (in the sense of Myers) is a double categorical yoga for describing the compositional structure of open dynamical systems. It unifies and improves on previous work on operadic notions of system theory, and provides a strong conceptual scaffolding for behavioral system theory. However, some of the most interesting systems out there escape the simple model of dynamical systems. They are instead cybernetic systems, or in other words, controllable dynamical systems. Notable and motivating examples are strategic games and machine learning models. In this talk I'm going to outline an upgrade of categorical system theory to deal with such systems by resorting to triple categories.

We improve the framework of open games with agency by showing how the players' counterfactual analysis giving rise to Nash equilibria can be described in the dynamics of the game itself (hence diegetically), getting rid of devices such as equilibrium predicates. This new approach overlaps almost completely with the way gradient-based learners are specified and trained. Indeed, we show feedback propagation in games can be seen as a form of backpropagation, with a crucial difference explaining the distinctive character of the phenomenology of non-cooperative games. We outline a functorial construction of arena of games, show players form a subsystem over it, and prove that their 'fixpoint behaviours' are Nash equilibria.

Twitter thread about it.

A talk about recent developments on dependent optics. A recording is available on YouTube.

Some developments on dependent optics, prompted by recent advances by Vertechi and Milewski. I obtained the same definition they proposed from a 'dependent' Tambara theory based on actions of double categories, but then shunned it away because I couldn't prove dependent lenses were an example. Vertechi found a way, Milewski found more examples, and thus I released my notes after some updating.

Twitter thread.

A long theory paper on actions of monoidal categories and their properties. We describe distributive laws between monoidal and actegorical structures and provide examples of their use in the theory of optics and parametric morphisms.

Twitter thread.

Using parametric dependent lenses for writing web servers.

Twitter thread.