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.”
David Jaz Myers
Fri Nov 17 2023
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”.
toby smithe
Fri Nov 17 2023
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…)
Matteo Capucci
Sun Nov 19 2023
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…
toby smithe
Sun Nov 19 2023
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!)
Matteo Capucci
Sun Nov 19 2023
Indeed, that one! (Here’s more characters to accompany my 1 bit of information reply)
toby smithe
Thu Nov 16 2023
Hmm, the abstract isn’t showing. Here it is:
The basic idea is simply “change of enrichment along a systems theory.”
David Jaz Myers
Fri Nov 17 2023
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”.
toby smithe
Fri Nov 17 2023
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…)
Matteo Capucci
Sun Nov 19 2023
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…
toby smithe
Sun Nov 19 2023
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!)
Matteo Capucci
Sun Nov 19 2023
Indeed, that one! (Here’s more characters to accompany my 1 bit of information reply)