Consider the category of groups with its forgetful functor down to Set. This functor is monadic, which tells us that groups are sets equipped with an algebraic structure. Now consider the category of finite sets, with its inclusion into Set. This functor is not monadic for many reasons, but what monadic functor best approximates it? In other words, finite sets are most like sets equipped with what algebraic structure? The answer to this question is given my the codensity monad of FinSet -> Set. Quite surprisingly, this is the ultrafilter monad, whose algebras are compact Hausdorff spaces!

That's all well an good, but why should you care? Every monad is trivially the codensity monad of some functor. Does it matter if it is the codensity monad of a specially nice one, say a fully faithful one? It does! For example, if a polynomial monad T on Set restricts to FinSet, then a general theory guarantees that there is a unique distributive law of the ultrafilter monad over T, i.e. a unique monad on CHaus which lifts T.

In this talk I will tell you about this recent development about distributive laws involving codensity monads, and how they relate to the main subject of my thesis: pushforward monads. We will use the ulftrafilter monad as our central example.