### 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 $T$, a formula for which is in the slides. What are the $T$-algebras?

I don’t know the answer.