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Stream: event: ACT20

Topic: July 6: Callum Reader's talk

Paolo Perrone (Jul 02 2020 at 02:35):

Hello all! This is the thread of discussion for the talk of Callum Reader, "Measures and Enriched Categories".
Date and time: Monday July 6, 17:10 UTC.
Zoom meeting: https://mit.zoom.us/j/7055345747
YouTube live stream: https://www.youtube.com/watch?v=1WhsWK20iRo&list=PLCOXjXDLt3pZDHGYOIqtg1m1lLOURjl1Q

Paolo Perrone (Jul 06 2020 at 17:01):

We start in 9 minutes!

Tomáš Jakl (Jul 06 2020 at 17:35):

:clap:

Paolo Perrone (Jul 06 2020 at 17:46):

Nice talk! Anybody else would be interested in looking at the extension of the Kantorovich monad to Lawvere metric spaces?

Mohamar Rios (Jul 06 2020 at 17:47):

I'm not sure if this is useful to you but you can get a topology on your space, X, with your pseudometric, d, by first defining balls as usual with the caveat that they may or may not be open
$B_\varepsilon(x)=\{y|d(x,y)<\varepsilon\}$
and then saying a set $U\subseteq X$ is open iff
$U=X$
or
$\forall u\in U, \exists \varepsilon>0, B_\varepsilon(u)\subseteq U$

I can link a citation if you like. I'm using this technique for an even more general thing called a prametric in my research so it should work for pseudometrics. The only thing I'm not 100% sure of right now is the infinite distance case because I handled that differently.

Edit: Sorry, I thought LaTeX worked on Zulip.

Paolo Perrone (Jul 06 2020 at 17:53):

It does, but you have to use two dollar signs!

Mohamar Rios (Jul 06 2020 at 17:55):

Thanks!

Paolo Perrone (Jul 07 2020 at 10:31):

Here's the video:
https://www.youtube.com/watch?v=8DWoH6Sw7ZI&list=PLCOXjXDLt3pYot9VNdLlZqGajHyZUywdI

Paolo Perrone (Jul 07 2020 at 12:07):

So here's what seems to be true (if I've understood the talk correctly): if by "measure" we mean a "functional on functions", then we can perfectly extend the Kantorovich metric to Lawvere metric spaces. The question is rather: is there a Riesz-style theorem for Lawvere metric spaces? In other words, what do those functionals correspond to?
And if there is one, do we also get a form of Kantorovich duality?