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.
Paolo Perrone (Jun 07 2020 at 17:48):Hey all,
During the last social event some of us mentioned filling the nLab with categorical probability. Let's do that!
Paolo Perrone (Jun 07 2020 at 18:01):Let's see what we can do more in practice. For what concerns classical probability (as opposed to quantum), there are three main related threads: Markov categories, probability monads, and valuations.
Paolo Perrone (Jun 07 2020 at 18:02):For what concerns probability monads, there is already some stuff, starting here (and links):
https://ncatlab.org/nlab/show/monads+of+probability%2C+measures%2C+and+valuations
Paolo Perrone (Jun 07 2020 at 18:02):See in particular the table at the end, https://ncatlab.org/nlab/show/monads+of+probability%2C+measures%2C+and+valuations#detailed_list
Paolo Perrone (Jun 07 2020 at 18:04):Things that could be done there are, first of all, expand the table with whatever monad is missing.
Also, following the links from the table, some content is missing or could be updated.
Moreover, something that's missing there is the duality part (think Riesz representation theorem).
Paolo Perrone (Jun 07 2020 at 18:04):Also, the article on the Giry monad could use a total makeover: https://ncatlab.org/nlab/show/Giry+monad
Paolo Perrone (Jun 07 2020 at 18:05):For valuations, there are for example these pages, https://ncatlab.org/nlab/show/valuation+%28measure+theory%29 and https://ncatlab.org/nlab/show/correspondence+between+measure+and+valuation+theory
Paolo Perrone (Jun 07 2020 at 18:06):What is missing there: locales, especially internal locales in a topos. That's something where I can't help (yet), since I'm not an expert. If someone could help, the topic would also be very much in the spirit of the nLab.
Paolo Perrone (Jun 07 2020 at 18:08):For Markov categories: these seem really to be the next big thing, if not the current. Let the nLab reflect that! For now, there is this (incomplete) article, https://ncatlab.org/nlab/show/Markov+category
Paolo Perrone (Jun 07 2020 at 18:08):Feel free to expand and create child articles as you see fit, and don't forget to add links and references ;)
Paolo Perrone (Jun 07 2020 at 18:09):Last but not least: traditional measure theory is missing. Recently some stuff has been added, but the concepts of random variables, joints and marginals, their laws, and so on, could use some expanding. Starting here: https://ncatlab.org/nlab/show/random+variable
Paolo Perrone (Jun 07 2020 at 18:10):I'm not an expert on quantum probability (yet?), so if anyone knows more, let's hear some suggestions.
Juan Pablo Vigneaux (Jun 08 2020 at 17:22):I propose to edit the article "probability theory", which is actually incomplete and extremely poor, to make some sort of "big index" of the different approaches to probability theory: summarize briefly the classical approach (this is already done in part), then briefly explain the categorical approaches and link to the main articles (valuations, Giry monad, Markov category, etc), and then add at the end diagrams and tables that would compare and relate this approaches.
Paolo Perrone (Jun 08 2020 at 17:55):Juan Pablo Vigneaux said:
I propose to edit the article "probability theory", which is actually incomplete and extremely poor, to make some sort of "big index" of the different approaches to probability theory: summarize briefly the classical approach (this is already done in part), then briefly explain the categorical approaches and link to the main articles (valuations, Giry monad, Markov category, etc), and then add at the end diagrams and tables that would compare and relate this approaches.
That sounds great. As a rule, before changing or removing existing content, it's always a good idea to check in with the community on the nForum.
Juan Pablo Vigneaux (Jun 08 2020 at 19:59):That sounds sensible. I've never edited the nLab.
I do not consider myself particularly competent to do what I proposed, but I could contribute with a part of it.
Paolo Perrone (Jun 08 2020 at 22:49):Here's something that we could do in practice: a big table of Markov categories with their properties.
For example: does the Kleisli category of the Giry monad on Polish spaces, as a Markov category, have conditionals?
One can look up this category in that table, and in the relevant entry, if the answer is known, there is a reference.
Paolo Perrone (Jun 09 2020 at 19:12):Paolo Perrone said:
Here's something that we could do in practice: a big table of Markov categories with their properties.
For example: does the Kleisli category of the Giry monad on Polish spaces, as a Markov category, have conditionals?
One can look up this category in that table, and in the relevant entry, if the answer is known, there is a reference.
I've started. Please contribute, whoever can. https://ncatlab.org/nlab/show/Markov+category#detailed_list