Codensity Monads
Posted by Tom Leinster
Yesterday I gave a seminar at the University of California, Riverside, through the magic of Skype. It was the first time I’ve given a talk sitting down, and only the second time I’ve done it in my socks.
The talk was on codensity monads, and that link takes you to the slides. I blogged about this subject lots of times in 2012 (1, 2, 3, 4), and my then-PhD student Tom Avery blogged about it too. In a nutshell, the message is:
Whenever you meet a functor, ask what its codensity monad is.
This should probably be drilled into learning category theorists as much as better-known principles like “whenever you meet a functor, ask what adjoints it has”. But codensity monads took longer to be discovered, and are saddled with a forbidding name — should we just call them “induced monads”?
In any case, following this principle quickly leads to many riches, of which my talk was intended to give a taste.
Posted at January 16, 2020 2:36 PM UTC
Re: Codensity Monads
Incidentally, here’s a puzzle I posed in the talk. The inclusion of the category of fields into the category of commutative rings has a codensity monad , a formula for which is in the slides. What are the -algebras?
I don’t know the answer.