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Given a set $X$, let $beta X$ denote the set of ultrafilters. The following theorems are known:

• 
  $X$ canonically embeds into $beta X$ (by taking principal ultrafilters);


• If $X$ is finite, then there are no non-principal ultrafilters, so $beta X = X$.


• If $X$ is infinite, then (assuming choice) we have $|beta X| = 2^2^X$.

These are reminiscent of similar claims that can be made about vector spaces and double duals:

• 
  $V$ canonically embeds into $V^star star$;


• If $V$ is finite-dimensional, then we have $V = V^star star$;


• If $V$ is infinite-dimensional, then (assuming choice) we have $dim(V^star star) = 2^2^dim(V)$.

This suggests that the operation of taking the collection of ultrafilters on a set can be viewed as a double iterate of some 'duality' of sets. Can this be made precise: that is to say, is there some notion of a 'dual' of a set $X$, $delta X$, such that the following are true?

• The double dual $delta delta X$ is (canonically isomorphic to) the set $beta X$ of ultrafilters on $X$;


• If $X$ is finite, then $|delta X| = |X|$ (but not canonically so);


• If $X$ is infinite, then (assuming choice) $|delta X| = 2^X$.

Apart from the tempting analogy between $beta X$ and $V^star star$, further evidence for this conjecture is that $beta$ can be given the structure of a monad (the 'ultrafilter monad'), and monads can be obtained from a pair of adjunctions.

This is a quite standard idea in functional analysis. Let $X$ be any set and let $c_0(X)$ be the space of all functions from $X$ to $mathbbC$ which go to zero at infinity. Then the algebra homomorphisms from $c_0(X)$ to $mathbbC$ are precisely the point evaluations at elements of $X$, i.e., the spectrum of $c_0(X)$ is naturally identified with $X$.

Going to the second dual we get $l^infty(X)$, the space of all bounded functions from $X$ to $mathbbC$, whose spectrum is naturally identified with $beta X$.

[deleted an additional comment which wasn't accurate]

This is an elaboration on Todd Trimble's comment about Tom Leinster's lovely posts about codensity monads. I quite like the codensity monad story; here is my preferred way of telling it.

Suppose you have a functor $F : C to D$. A general question to ask about it is this:

What additional structure, beyond being objects in $D$, do the objects $F(c) in D$ canonically have, by virtue of having been spit out by $F$?

A simple construction is that the objects $F(c)$ canonically admit an action by the automorphism group $textAut(F)$ of $F$ as a functor, more or less by definition, and more generally by the endomorphism monoid of $F$. This observation can already be used to motivate Weyl groups and Hecke algebras.

A more elaborate construction is that if $F$ admits a left adjoint $G : D to C$, then the objects $F(c)$ canonically admit an action by the monad $T = FG : D to D$, by which I mean they are canonically algebras over this monad. In nice cases (see monadic adjunction and monadicity theorem) this completely characterizes $C$ in terms of $D$ and $T$, for example if $D = textSet$ and $C$ is a typical algebraic category such as groups, rings, modules. A more unusual example here is that $C$ can be compact Hausdorff spaces, and then $T$ is the ultrafilter monad.

But there's an even more general construction than this, which can be motivated in several ways. Here's one. Suppose a monoidal category $M$ acts by endomorphisms on a category $E$, meaning we have a monoidal functor $M to [E, E]$, where $[E, E]$ is the monoidal category of endofunctors $E to E$. This is the minimal setup we need to talk about a monoid $m in M$ acting on an object $e in E$; see this blog post where I use this setup to motivate the definition of a monad.

Now, given an object $e in E$, we can ask for the universal monoid in $M$ which acts on $e$, which is an "$M$-internal" notion of the endomorphism monoid of $e$. This monoid $m in M$, if it exists, is defined by the universal property that maps $n to m$ of monoids are in natural bijection with actions of $n$ on $e$. If $M = [E, E]$, then this construction, when it exists, recovers the endomorphism monad of $e$. If $E = M$ acting on itself by left multiplication, then this construction, when it exists, recovers the internal endomorphism object of $e$.

In our setting we want to apply this construction to $E = [C, D]$ and $M = [D, D]$, where $[D, D]$ acts on $[C, D]$ by postcomposition. That is, we want a monad $T : D to D$ which universally acts on a functor $F : C to D$ in the sense that maps of monads to $T$ are in natural bijection with actions of monads on $F$.

Claim: This monad, if it exists, is the codensity monad of $F$.

(I don't have a reference for this, although it's closely related to the definition of the codensity monad as the right Kan extension of $F$ along itself; I remember convincing myself of this a few years ago, around the time I wrote this blog post on monads, and then I never wrote up the details. Welp.)

Now the really fun fact, which Todd Trimble alludes to above, is:

The codensity monad of the inclusion $textFinSet to textSet$ is the ultrafilter monad, and the codensity monad of the inclusion $textFinVect to textVect$ is the double dual monad.

This sets up a lovely analogy between compact Hausdorff spaces (algebras over the ultrafilter monad) and whatever algebras over the double dual monad are; Tom and Todd call them "linearly compact vector spaces" but my preferred terminology here is just "profinite vector spaces," in that the category is precisely $textPro(textFinVect)$.