Relational $\beta$-modules

Idea

One of my early Honours students at Macquarie University? baffled his proposed Queensland graduate studies supervisor who asked whether the student knew the definition of a topological space. The aspiring researcher on dynamical systems answered positively: “Yes, it is a relational $\beta$-module!” I received quite a bit of flak from colleagues concerning that one; but the student Peter Kloeden? went on to become a full professor of mathematics in Australia then Germany.

—Ross Street, in An Australian conspectus of higher categories?

In 1970, Michael Barr gave an abstract definition of topological space based on a notion of convergence between ultrafilters (building on work by Ernest Manes on compact Hausdorff spaces). Succinctly, Barr defined topological spaces as ‘relational $\beta$-modules’. It was subsequently realized that this was a special case of the notion of generalized multicategory. Here we unpack this definition and examine its properties.

The correctness of this definition (in the sense of matching Bourbaki's definition) is equivalent to the ultrafilter principle ($\mathrm{UP}$). However, the definition can be treated on its own, even in a context without $\mathrm{UP}$. So we also consider the properties of relational $\beta$-modules when these might not match Bourbaki spaces.

Definitions

Abstract

If $S$ is a set, let $\beta S$ be the set of ultrafilters on $S$. This operation $\beta$ extends to a functor on Set and even on Rel as follows: Given a binary relation $R:X\to Y$ and ultrafilters $𝒰$ on $X$ and $𝒱$ on $Y$, we have $𝒰$ related to $𝒱$ via $\beta (R)$ iff …

This functor $\beta$ becomes a monad on $\mathrm{Rel}$ and even on $\mathrm{Set}$ as follows: …

Then a relational $\beta$-module is a lax algebra? (module) of $\beta$ on the 2-poset $\mathrm{Rel}$.

Arguably, it is better to consider $\mathrm{Rel}$ as a proarrow equipment in this construction; otherwise it is difficult to extract the notion of continuous function between topological spaces. See generalized multicategory.

Concrete

A relational $\beta$-module is a set $S$ and a binary relation $\to$ between ultrafilters on $S$ and elements of $S$ that satisfies a certain rule.

To explain this rule, we first define a subset $A$ of $S$ to be open if $A\in 𝒰$ whenever $𝒰\to x\in A$. Then the requirement on $\to$ is a converse:

• $𝒰\to x$ if $A\in 𝒰$ whenever $A$ is open and $x\in A$.

…

This definition exhibits a topological space as a particular type of pseudotopological space. (A pseudotopological space is just a relational $\beta$-module which omits the “associativity” axiom.)

Properties

We say that $𝒰$ converges to $x$ if $𝒰\to x$.

As above, a subset $A$ of $S$ is open if $A\in 𝒰$ whenever $𝒰\to x\in A$. On the other hand, $A$ is closed if $x\in A$ whenever $A\in 𝒰\to x$.

A relational $\beta$-module is compact if every ultrafilter converges to at least one point. It is Hausdorff if every ultrafilter converges to at most one point. Thus, a compactum is (assuming $\mathrm{UF}$) precisely a relational $\beta$-module in which every ultrafilter converges to exactly one point, that is in which the action of the monad $\beta$ lives in $\mathrm{Set}$ rather than in $\mathrm{Rel}$; see ultrafilter monad.

…

Relation to nonstandard analysis

In nonstandard analysis (which implicitly relies throughout on $\mathrm{UF}$), one may define a topological space using a relation between hyperpoints (elements of $S^{*}$) and standard points (elements of $S$). If $u$ is a hyperpoint and $x$ is a standard point, then we write $u\approx x$ and say that $x$ is a standard part? of $u$ or that $u$ belongs to the halo? (or monad, but not the category-theoretic kind) of $x$. This relation must satisfy a condition analogous to the condition in the definition of a relational $\beta$-module. The nonstandard defintions of open set, compact space, etc are also analogous. (Accordingly, one can speak of the standard part of $u$ only for Hausdorff spaces.)

So ultrafilters behave very much like hyperpoints. This is not to say that ultrafilters are (or even can be) hyperpoints, as they don't obey the transfer principle?. Nevertheless, one does use ultrafilters to construct the models of nonstandard analysis in which hyperpoints actually live. Intuitions developed for nonstandard analysis can profitably be applied to ultrafilters, but the transfer principle is not valid in proofs.

Relation to other topological concepts

If $\beta$ is treated as a monad on $\mathrm{Set}$ instead of on $\mathrm{Rel}$, then its algebras are the compacta (the compact Hausdorff spaces), again assuming $\mathrm{UF}$; see ultrafilter monad.

One might hope that there would be an analogous treatment of uniform spaces based on an equivalence relation between ultrafilters. (In nonstandard analysis, this becomes a relation $\approx$ of infinite closeness between arbitrary hyperpoints, instead of only a relation between hyperpoints and standard points.) The description in terms of generalized multicategories is known to generalize to a description of uniform spaces, but rather than using relations between ultrafilters, this description uses pro-relations between points.