# Ultrafilters as a double dual

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.

• Well, one of the very first things that comes to mind that's sort of in this vein is that $\beta X = \hom_{\text{Bool}}(\hom_{\text{Set}}(X, 2), 2)$. But if you want to pursue your analogy at a deeper level, try golem.ph.utexas.edu/category/2012/09/…, where both the ultrafilter monad and the double dualization monad are reckoned to be codensity monads induced by the full inclusions of finitary objects. – Todd Trimble Mar 7 '19 at 15:43
• Close to Todd's comment, I'd view $\beta X$ as $F(X)=\mathrm{hom}_{\mathrm{Bool}}(\mathrm{hom}_{\mathrm{Top}}(X,\mathbf{Z}/2\mathbf{Z}))$. In general, I guess that for a topological space $X$, the map $X\to F(X)$ is the initial object for the category of continuous maps from $X$ to compact Hausdorff totally disconnected topological spaces. A difference with taking biduals is that $F(F(X))=F(X)$ by Stone duality. – YCor Mar 7 '19 at 16:17
• Well, one major difference is that without choice it is always the case that $X$ embeds into $\beta X$, it's just not provable that the embedding is not surjective; whereas $V^*$ might be trivial, let alone $V^{**}$, even though $V$ isn't. – Asaf Karagila Mar 7 '19 at 16:30
• (My point above, is that the canonical embedding of $V$ into $V^{**}$ uses choice in a subtle way, whereas the canonical embedding of $X$ into $\beta X$ does not.) – Asaf Karagila Mar 8 '19 at 8:57
• @AsafKaragila Interesting, never thought of it - can one prove anything about the kernel of $V\to V^{**}$ without choice? Could you recommend a text about those things? – მამუკა ჯიბლაძე Mar 8 '19 at 11:55

## 2 Answers

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 $$\mathbb{C}$$ which go to zero at infinity. Then the algebra homomorphisms from $$c_0(X)$$ to $$\mathbb{C}$$ 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 $$\mathbb{C}$$, whose spectrum is naturally identified with $$\beta X$$.

[deleted an additional comment which wasn't accurate]

• What does it mean for a function from a set $X$ to $\mathbb{C}$ to "go to zero at infinity"? – Alex Kruckman Mar 7 '19 at 19:28
• For any $\epsilon > 0$, there is a finite subset of $X$ off of which $|f(x)| \leq \epsilon$. – Nik Weaver Mar 7 '19 at 19:36
• Equivalently, the extension of $f$ to the one point compactification of $X$ (with the discrete topology) which sets $f(\infty) = 0$ is continuous. Hence "goes to zero at infinity". – Nik Weaver Mar 7 '19 at 19:38
• @NikWeaver Elegant! This essentially answers both of my questions, namely why the analogy exists, and also why the 'half-iterate' $\delta X$ cannot be defined: when you take the first dual of $c_0(X)$, the result is not a C*-algebra, so you can't take its spectrum. But the double dual $l^{\infty}(X)$ is a C*-algebra, so you can indeed take its spectrum, and you get $\beta X$. – Adam P. Goucher Mar 8 '19 at 14:57
• Yes, that's right. The first duals generally don't even have a preferred product, so there's nothing like a spectrum. – Nik Weaver Mar 8 '19 at 16:25

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 $$\text{Aut}(F)$$ 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 $$D = \text{Set}$$ 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 $$\text{FinSet} \to \text{Set}$$ is the ultrafilter monad, and the codensity monad of the inclusion $$\text{FinVect} \to \text{Vect}$$ 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 $$\text{Pro}(\text{FinVect})$$.

• There is a kind of "double dual" in this story, by the way: one way to describe the codensity monad of $F : C \to D$, if $C$ is essentially small and $D$ has small limits, is that it's the monad associated to the adjunction between $D$ and $[C, \text{Set}]^{op}$ whose left adjoint sends $d \in D$ to the functor $\text{Hom}(d, F(-)) : C \to \text{Set}$. When $F$ is the inclusion of finite sets into sets this is a disguised form of $\beta X$ and when $F$ is the inclusion of finite-dimensional vector spaces into vector spaces this is a disguised form of taking the dual. – Qiaochu Yuan Mar 9 '19 at 20:39
• (Then the right adjoint is the "second dual," although it's a bit trickier to describe.) – Qiaochu Yuan Mar 9 '19 at 20:42