Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: event: ACT20

Topic: July 6: Jules Hedges' talk

Paolo Perrone (Jul 02 2020 at 02:39):

Hello all! This is the thread of discussion for the talk of Jules Hedges, "Non-compositionality in categorical systems theory".
Date and time: Monday July 6, 12:10 UTC.
Zoom meeting: https://mit.zoom.us/j/7055345747
YouTube live stream: https://www.youtube.com/watch?v=1A76CO3K28U&list=PLCOXjXDLt3pZDHGYOIqtg1m1lLOURjl1Q

Jules Hedges (Jul 06 2020 at 10:38):

My slides: https://obsoletewallstreet.files.wordpress.com/2020/07/categorical_systems_theory.pdf

Paolo Perrone (Jul 06 2020 at 12:08):

We start in 2 minutes!

Martti Karvonen (Jul 06 2020 at 12:33):

I'm wondering what guarantees that the set of comorphisms is a set rather than a proper class (unless your category is small)? Is some quotioning along some equivalence relation on contexts doing the work in the background?

Sophie Libkind (Jul 06 2020 at 16:47):

Really enjoyed your talk, Jules! I'm wondering about a possible example. In Compositional Framework for Reaction Networks they define a black-boxing functor out of the category of continuous open dynamical systems. This black-box is trying to capture the behavior of "steady states." Unfortunately, pure steady states is only lax functorial and there may be emergent steady states. As a solution to making the black-box functorial, the behavior they choose is instead "steady states and a flow on the boundary which keeps the state steady." I wonder if this "flow on the boundary" is an example of a context?

Jules Hedges (Jul 06 2020 at 16:50):

Oo, nice idea, I'll definitely look into that

Jules Hedges (Jul 06 2020 at 16:50):

I'm on the hunt for a collection of nice examples, until then I have no real reason to write a paper about this...

Sophie Libkind (Jul 06 2020 at 16:51):

I've actually been thinking about generalizing this exact black boxing functor but as a transformation between operad algebras

Sophie Libkind (Jul 06 2020 at 16:51):

Does the context formalism generalize to operad algebras?

Jules Hedges (Jul 06 2020 at 16:53):

I think it should. Working out the conditions it should satisfy might take some work

Toby Smithe (Jul 06 2020 at 17:00):

Hey guys! I have also been precisely thinking about this problem lately (and @Jules Hedges I was meaning to e-mail you and @Eigil Rischel about this because I think my systems are related to the discussion I saw between you on Twitter). This 'flow on the boundary' thing is (I think) very closely related to something I briefly e-mailed @Sophie Libkind about, with respect to Lagrangian dynamics; and in this case, the composition of my systems seems very clearly operadic (roughly, the physical intuition is like "particles within particles"). I'll write up some notes on my system and send them round :-)

Eigil Rischel (Jul 06 2020 at 17:03):

I guess you want a set of contexts for every type (i.e color) in your operad, which is in some sense "reverse functorial" in the algebra describing your systems - i.e if you have an operad operation $o: (a,b) \to c$, a context of type $c$ and a system of type $a$, you obtain a context of type $b$

Eigil Rischel (Jul 06 2020 at 17:13):

We can take a coalgebra of our operad, by which I mean an algbra in $(Set^{op},\times)$. This means there's a map from $Ctx(c) \to Ctx(a,b)$ in the situation above. Then maybe pasting a system into a context is some sort of action of the algebra of systems on this gadget?

Sophie Libkind (Jul 06 2020 at 17:30):

What does it mean to have a coalgebra of an operad? And this coalgebra would be the contexts $Ctx$?

Eigil Rischel (Jul 06 2020 at 17:33):

Yeah, by that I meant an algebra in an opposite category of sets (but still with cartesian product as monoidal structure). But now that you mention it, this doesn't seem quite right - given an operation $o(a,b) \to c$, and a context of type $c$, I don't expect to find two contexts of type $a$ and type $b$.. so something's off here.

Paolo Perrone (Jul 06 2020 at 20:56):

Here's the video!
https://www.youtube.com/watch?v=JxvmjOZbv7U&list=PLCOXjXDLt3pYot9VNdLlZqGajHyZUywdI