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Stream: learning: questions

Topic: Duality in categorical systems theory

David Corfield (Jul 29 2026 at 11:25):

Categorical systems theory comes into close contact with control theory and automata theory, and it's not long venturing into this world before one encounters the pervasive duality between controllability/reachability and observability. This is very much the terrain of Kalman (from around 1960) on linear systems duality, and his work with Arbib and others (from around 1970)) providing a unified account of systems and automata. I began to sketch an nLab page about the duality at [[controllability and observability]], but there's far more to include there.

It's an easy step then to understand instances of this duality as a form of Chu duality. (Here is a sketch of why Kalman duality is a form of Chu duality.)

Since we all love duality in the CT community, one might expect to see controllability-observability duality as a prominent topic in ACT approaches to systems theory. For instance, it's present in Lyapunov theory. But outside of work on minimization of automata, I'm seeing very little. Am I missing it?

Jules Hedges (Jul 29 2026 at 11:28):

There's some things about Kalman duality in the thesis of my Phd student Riu Rodríguez Sakamoto, I should check to see whether that's become publicaly accessible yet

Jules Hedges (Jul 29 2026 at 11:29):

https://stax.strath.ac.uk/concern/theses/r207tq035

Jules Hedges (Jul 29 2026 at 11:38):

Things got a bit messy when we got into the real details, not as categorically clean as I was hoping it would be. I bet there is much more to say about this topic

Jules Hedges (Jul 29 2026 at 11:40):

Perhaps @Riu Rodríguez Sakamoto will receive a ping from this

David Corfield (Jul 29 2026 at 15:21):

I see Riu has spoken on Decorated Para for linear quadratic regulators, right at the heart of optimal control-optimal estimator duality. Here the duality with the Kalman filter.

David Corfield (Jul 31 2026 at 15:28):

Did anyone working on polynomial functors consider $Chu(Poly, y)$, or perhaps $Chu(Poly, Ry)$ for some value object $R$? One might consider Interaction laws of monads and comonads for this.

Bob Atkey (Jul 31 2026 at 17:04):

I've thought about $\mathrm{Chu}(\mathrm{Poly}, y)$ as a proof relevant model of the BV logic, where sequencing is interpreted as the composition product. One compelling thing is that a "Kleene star" like construct would be modelled using the ideas in Libkind and Spivak's Pattern Runs on Matter paper.

David Corfield (Aug 04 2026 at 10:52):

Thanks, @Bob Atkey! Seems like there are some literatures to link up usefully.

David Corfield (Aug 05 2026 at 11:31):

Bob Atkey said:

...modelled using the ideas in Libkind and Spivak's Pattern Runs on Matter paper.

Do you mean for say $(p, q, \phi: p \otimes q \to y)$ in $Chu(Poly, y)$, something they might put as $\mathfrak{m}_p \otimes \mathfrak{c}_q \to \mathfrak{m}_{p \otimes q} \to \mathfrak{m}_y \cong \mathbb{N}y$?

David Corfield (Aug 05 2026 at 12:38):

One might also try $Ry$ instead of $y$ for some monoid $R$. Then $\mathfrak{m}_{Ry} \cong R^{\ast}y$ and there would be a multiplication map to $Ry$.