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.
John van de Wetering (Nov 25 2020 at 12:26):I would like some opinions on a result I have, as I am not sure whether it is interesting or new enough to actually write down and try to get published.
John van de Wetering (Nov 25 2020 at 12:27):It concerns the categorical motivation behind the real unit interval . The unit interval is of course of fundamental importance to probability theory and for instance homotopy theory, but at first glance it seems categorically quite arbitrary.
John van de Wetering (Nov 25 2020 at 12:29):My result is that there is a monad on the category of bounded posets (posets which have a minimum and maximum), of which the unit interval is an object in the Eilenberg-Moore category. This EM category has products, and the unit interval is 'irreducible' with respect to products. It is only one of three objects in this category that are irreducible. The others are the initial object consisting of the Booleans , and the other is the initial object .
John van de Wetering (Nov 25 2020 at 12:30):Hence, the singleton, the Booleans and the real unit interval are special because they are the unique irreducible objects in this Eilenberg-Moore category of a monad on the very simple category of bounded posets.
John van de Wetering (Nov 25 2020 at 12:33):Of course, this wouldn't mean much if the monad directly referred to real numbers itself. The EM category is actually equivalent to the category of omega-complete effect monoids, a quite nice abstract algebraic structure. I actually start with this category and show it is an EM category using Beck's monadicity theorem, which admittedly does hurt the niceness of the construction.
Nathanael Arkor (Nov 25 2020 at 12:43):(I don't have anything useful to say about your result, but let me just point out Freyd's Algebraic real analysis, which also considers how [0, 1] arises categorically in a different, coalgebraic sense, just in case you haven't come across it.)
John van de Wetering (Nov 25 2020 at 16:18):Thanks for the reference. I was sure there must be similar stuff to my result, and this indeed seems to be quite in that ballpark.
Dan Doel (Nov 25 2020 at 17:57):There was a talk recently at the NY Category Theory Seminar about the coalgebraic stuff.
