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 Baez (Feb 21 2023 at 19:54):A long time ago, my friend James Dolan explained that 'carrying' - that thing you do when you add numbers - is an example of what mathematicians call a '2-cocycle'. This just means that since addition is associative, carrying has to obey some equation.
So, you first met 2-cocycle in elementary school!
You can now read his expository posts on this in a nice format here:
thanks to Timothy Chow.
John Baez (Feb 21 2023 at 19:55):
Steve Huntsman (Feb 21 2023 at 20:14):John Baez said:
So, you first met 2-cocycle in elementary school!
See also A Cohomological Viewpoint on Elementary School Arithmetic
John Baez (Feb 21 2023 at 21:26):Yes! If you look carefully you'll see the author of that paper credits James Dolan for the idea.
John Baez (Feb 21 2023 at 21:26):Timothy Chow LaTeXed Dolan's posts because he wanted him to get more credit for the idea.
Keith Elliott Peterson (Feb 22 2023 at 00:59):Reminds me of the time John said something along the lines that most "complex" ideas in math can be explained easily.
John Baez (Feb 22 2023 at 01:03):The truly most complex ideas are specialized things like the classification of finite simple groups. I don't think these can be explained simply. But the ideas that are the most important and general tend to be fairly simple - that's why they're important and general.
John Baez (Feb 22 2023 at 01:05):By analogy: if there were a tool roughly like a hammer but it was extremely complex and took a huge amount of training to use, it wouldn't be much good.
dusko (Feb 22 2023 at 02:00):this makes a state machine where 2-cocycles are the states and 1-cochains the transitions... very very beautiful!
:pray: for sharing!
dusko (Feb 22 2023 at 02:22):do we know what would a similar account of multiplication look like? there also begins with a table and carrying, but there is the 2nd order carrying of the intermediary results, usually as external states. in the end we apply the addition. is it the direct product of the two exact sequences?
Mike Shulman (Feb 22 2023 at 05:38):John, why does the link go to archive.org instead of directly to the URL?
John Baez (Feb 22 2023 at 06:52):Because I predict archive.org will last longer than Timothy Chow's website. Also, Chow was the one who gave me that archive link.
David Michael Roberts (Feb 22 2023 at 10:34):Chow has experience from rotting links on MathOverflow...
John Baez (Feb 22 2023 at 18:32):dusko said:
:pray: for sharing!
Thanks! By the way, I know I owe you an email on another topic.
John Baez (Feb 22 2023 at 18:32):dusko said:
do we know what would a similar account of multiplication look like? there also begins with a table and carrying, but there is the 2nd order carrying of the intermediary results, usually as external states. in the end we apply the addition. is it the direct product of the two exact sequences?
It would take me some time to figure out the similar account for multiplication. The account for groups involves how central extensions of groups are described by 2-cocycles in group cohomology; I think there's a similar story for rings but I don't know it as well!
Basically, all the "carrying" information required to build out of two copies of should be accounted for by some 2-cocycle. And similarly for building out of and another copy of .