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.
I saw a paper of @Jonathan Gorard on the arXiv presenting Categorica, built on top of Mathematica. Has anyone tried playing with it?
No, but I want to. Is it proprietary?
There is a GitHub repository: https://github.com/JonathanGorard/Categorica
However, it does not contain a license, which AFAIU means that it has the default strong copyright protections. And, of course, Mathematica is itself proprietary.