Animating categories

This is a companion discussion topic for the original entry at

Hmm, the abstract isn’t showing. Here it is:

It can sometimes be helpful to animate categories, to render their morphisms dynamical. This article explains how, given a systems theory and a suitably-enriched category, the latter may be animated by the former, yielding a bicategory whose 1-cells are systems that “emit morphisms”. The construction is exemplified by cilia: dynamical systems that control lenses. A further step of enrichment then yields, and generalizes, the “dynamic categories” of Shapiro and Spivak (2023).

The basic idea is simply “change of enrichment along a systems theory.”

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I’m going to read this when I can scrape a free moment, but I just wanted to say that I love the name “cilia”.

Thanks, David! It looks long, but it’s mostly spelling out examples: as I say, the basic idea is pretty simple. So with your background you shouldn’t need more than a moment.

(Really I wrote it to make it easier/neater to explain the compositional structure of Bayesian predictive coding in a follow-up…)

I wonder if animation and ‘dependent’ Para construction are related. To start: is there a way to see enrichment as a fibred action, or viceversa?

This question goes back three years at least, where we trying to relate ‘external Para’ (working by means of enrichment) and ‘internal Para’ (working by actegories). We knew both where equivalent in the tensored case. I think recently Dylan Braithwaite found a way to also describe them both in a unified way without tensoring.
Perhaps one can do a similar trick here…

That’s a nice question and it would make things a bit neater if so… I’m not quite satisfied with the notion of “indexed enrichment” used there. I shall meditate on it.

Meanwhile, by Dylan’s unification, do you mean using locally graded categories? (I can almost see how that might go, having spent maybe 15 seconds pondering it!)

Indeed, that one! (Here’s more characters to accompany my 1 bit of information reply)