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 -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 -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 (). However, the definition can be treated on its own, even in a context without . So we also consider the properties of relational -modules when these might not match Bourbaki spaces.
we define to be the relation obtained by taking the image of . It turns out, although it is by no means obvious, that is according to this definition a strict functor on .
The monad structure on , given by a unit and multiplication , extends not to a strict monad on , but rather one where the transformations are op-lax in the sense of there being inequalities
(while of course the monad associativity and unit conditions remain as equations: hold on the nose).
Arguably, it is better to consider as a proarrow equipment in this construction, in order to accommodate continuous functions between topological spaces (not continuous relations!) as the appropriate abstract notion of morphism between relational -modules. We touch on this below, but for a much wider context, see generalized multicategory.
A relational -module is a set and a binary relation between ultrafilters on and elements of that satisfy the conditions (1) . For and , we write if satisfies the relation , or often just if the relation is clear. We pronounce this by saying “the ultrafilter converges to the point ”, so that plays the role of “notion of convergence”.
Preliminary to explaining the conditions (1), we first set up a Galois connection between and subsets , so that fixed points on the side are exactly topologies on , and fixed points on the other side are (as we show below) lax -module structures on . The Galois connection would then of course restrict to a Galois correspondence between topologies and lax module structures.
Recall that each topology induces a notion of convergence where means ( contains the filter of neighborhoods of ). Accordingly, for general , define the relation by
Conversely, a topology can be retrieved from its notion of convergence: under the ultrafilter principle, the neighborhood filter of a point is just the intersection of all ultrafilters containing it (hence all such that ), and then a set is open if it is a neighborhood of all of its elements. Accordingly, for general “notions of convergence” , we define a collection by
is a topology on , for any .
It is trivial that . If , and if and , then also and and we conclude , whence since is an ultrafilter, so that satisfies the condition of belonging to . Given a collection of elements , if and , then for some and we conclude , whence since is upward closed. Therefore satisfies the condition of belonging to (vacuously so if the collection is empty).
There is a Galois connection between notions of convergence on and subsets of , according to the bi-implication
To establish the bi-implication, it suffices to observe that both containments and are equivalent to the condition
If is a topology on , then (i.e., topologies are fixed points of the closure operator ).
We already have from Proposition 2. For the other direction, we must show that any belonging to is an -neighborhood of each of its points. Suppose the contrary: that but is not an -neighborhood of . Then for every -neighborhood , we have , so that sets of this form generate a filter. By the ultrafilter principle, we may extend this filter to an ultrafilter ; clearly we have and , but since and and , we also have , which is inconsistent with .
A pseudotopological space is set equipped with a relation such that .
All that remains is to check is:
The lax unit condition holds if , for a topology .
The unit may also be denoted , as it takes an element to the principal ultrafilter
and now the unit condition says for all . For , this says , or that for all neighborhoods , which is a tautology.
One of our goals is to prove the following theorem:
(Main Theorem) An arrow in is of the form if and only if the following inequalities are satisfied:
where is the multiplication on the ultrafilter monad.
Before rolling up our sleeves and proving the main theorem, we pause to consider some more abstract contexts in which to place the concept of lax -module, leading up to the context of generalized multicategories.
First we examine more closely the extension of the ultrafilter functor to , showing in particular that the extension is a strict functor. First we slightly rephrase our earlier definition:
For a relation between sets, given by a subobject in with projections and , define to be the composite
in the bicategory of relations.
Any span of functions that represents (in the sense that in the bicategory of relations) would serve in place of , since for any such span there is an epi with , , whence is epi (because the epi splits in ) and we have
In particular, is well-defined. Since extends , there is no harm in writing in place of . If , then (as can be seen from the calculation displayed above, but replacing the epi by a general map , and the first equation by an inequality ).
The same recipe works to extend any functor to , and the extension is always an op-lax functor in the sense that
as is easily seen by contemplating a pullback diagram (where and ):
whereupon one calculates
where the inequality comes from , which is equivalent to (where even equality holds). This calculation shows that is an actual (not just an op-lax) functor on iff satisfies the Beck-Chevalley condition: if is a pullback of , then
This in turn amounts to preserving weak pullbacks. (It actually says takes pullbacks to weak pullbacks, but this implies takes weak pullbacks to weak pullbacks because any endofunctor on preserves epis, using the axiom of choice.)
The functor satisfies the Beck-Chevalley condition (and therefore the extension is a strict functor).
Referring to the pullback diagram in Remark 1, let be the pullback. We must show that the canonical map
has a dense image. Let , so that are the same ultrafilter . Let and be basic open neighborhoods of and in and respectively; we must show that there is such that
or in other words such that and . We have since and , so that belongs to
and similarly . It follows that so that . Any element can be written as and for some and , and this completes the proof.
As mentioned in an earlier section, the natural transformations , do not extend to (strict) natural transformations on the locally posetal bicategory , but only to transformations that are op-lax in the sense of inequalities
for every relation . These are equivalent to inequalities
and they may be deduced simply by staring at naturality diagrams in , in which we represent or tabulate by :
To get an actual monad, it is more satisfactory in this context to consider not the bicategory , but rather the equipment or framed bicategory . That is, there is a 2-category of equipments (as a sub-2-category of a 2-category of double categories), so that the notion of monad makes sense therein, and it turns out the data to hand induces such a monad .
In more detail: the 0-cells of are sets, and the horizontal arrows are relations between sets. Vertical arrows are functions between sets, and a 2-cell of shape
is an inequality . We straightforwardly get a double category , and the ultrafilter functor on extends to a functor between double categories (or in this case, equipments), preserving all structure in sight.
Some attention must be paid to the notion of transformation between functors between equipments. A transformation assigns to each 0-cell of a vertical arrow , and to each horizontal arrow a 2-cell of the form
suitably compatible with the double category structures.
We thus find that the op-lax structures of the transformations , on qua bicategory are exactly what we need to produce honest transformations , on qua equipment, and the result is an ultrafunctor monad on the equipment .
Given a monad on an equipment , one may proceed to construct a horizontal Kleisli equipment with the same 0-cells and vertical arrows as , but whose horizontal arrows are of the form . A 2-cell in (with vertical source and vertical target ) is a 2-cell in of the form
with horizontal compositions being performed in familiar Kleisli fashion. (When we say “familiar Kleisli fashion”, we are using the fact that an equipment allows one to “translate” vertical arrows, in particular the map , into horizontal arrows, which are then composed horizontally. Similarly, the unit of the monad is translated into a horizontal arrow, where it plays the role of an identity in the Kleisli construction.)
In an equipment, there is a notion of monoid and monoid homomorphism. A monoid consists of a horizontal arrow together with unit and multiplication 2-cells
satisfying evident identities. A monoid homomorphism from to consists of a vertical arrow and 2-cell of the form
that is suitably compatible with the unit and multiplication cells.
The following notion gives an interim notion of generalized multicategory that applies in particular to relational -modules.
Given a monad on an equipment , a -monoid is a monoid in the horizontal Kleisli equipment . A map of -monoids is a homomorphism between monoids in .
For the ultrafilter monad on the equipment , a structure of -monoid is equivalent to a structure of relational -module, and a homomorphism of -monoids is the same as a lax map of relational -modules in the bicategory .
This is really just a matter of unwinding definitions. The data of a -monoid in the equipment amounts to a set together with a horizontal arrow in the Kleisli construction, that is to say a relation (opposite to our conventional direction, i.e., ). The unit and multiplication cells for are inequalities and (the vertical source and target being identity maps), where the identity in the Kleisli construction uses the unit for and the Kleisli composition uses the multiplication. Back in the bicategory these translate to relational inequalities
or, with ,
These boil down to relational inequalities
as in the axioms on relational beta-modules. Similarly, a -monoid homomorphism is a vertical arrow in together with a suitable 2-cell, which after some unraveling comes down to a relational inequality
or to an inequality , which may be further massaged into the form , or simply to
as advertised in the notion of lax morphism of relational -modules (cf. theorem 4 below).
In some sense, relational -modules as presented here are a toy example of generalized multicategory theory as set out by Cruttwell and Shulman, where they argue that in order to get a fully satisfying theory that unifies all the relevant constructions and examples, one should really work in the context of monads acting on virtual equipments and study normalized -monoids. Explaining all this requires a lengthy build-up. Even in the relatively restricted packet of unifications that come under the rubric of -algebras, as studied by Clementino, Hofmann, Tholen, Seal and others as a way of bringing topological spaces, uniform spaces, metric spaces, approach spaces, closure spaces, and related notions under one conceptual umbrella, the relevant constructions (e.g. of canonical and op-canonical extensions of taut monads to lax monads on -matrices) can be somewhat elaborate, and mildly daunting.
The example of the ultrafilter monad acting on has just enough niceness to it (e.g., the Beck-Chevalley condition) that we are able to elide over most of the complications, while still giving a taste of the generality that goes beyond the “classical” examples of generalized multicategories involving cartesian monads and Kleisli constructions on bicategories of spans. Thus, relational beta-modules can serve as a useful key of entry into this subject.
We now return to the task of proving theorem 1.
For a topological space and a point , let , and let be the collection of open neighborhoods of . Then , i.e., , is equivalent to . This is because is the filter generated by .
The following conditions are equivalent:
for some ;
for some topology ;
Indeed, by general properties of Galois connections. Applying to both sides of the first equation, we have
so the first equation implies the third. Which in turn implies the second, since we know by proposition 1 that collections of the form are topologies. The second equation trivially implies the first.
We now break up our Main Theorem 1 into the following two theorems.
If for a topology , then the two inequalities of (1) are satisfied.
The first inequality (lax unit condition) was already verified in proposition 4. For the second (lax associativity), let us represent the relation by a span , so that . The lax associativity condition becomes
which (using ) is equivalent to
or in other words that for all ,
Here is, by definition,
with defined similarly. The monad multiplication is by definition
where (see also the previous section).
This would naturally follow if
But a pair belongs to if and ; we want to show this implies belongs to , or in other words that . But this is tautological, given how is defined in terms of a topology .
The next theorem establishes the converse of the preceding theorem; the two theorems together establish the Main Theorem. First we need a lemma.
Given any relation and , , we have that belongs to the closure wrt the topology if and only if .
As usual, let denote the complement of a subset . By definition of the topology , we have that is a neighborhood of if . In other words,
since in an ultrafilter , we have iff . Negating both sides of this bi-implication gives
If in satisfies the inequalities of (1), then .
If , then every neighborhood of belongs to , so that for every , every neighborhood of intersects in a nonempty set. But this just means for every , or in other words (using lemma 1) that
Representing the relation as usual by a subset , another way of expressing the existential formula on the right of this entailment is:
or even just
as subsets of , as ranges over all elements of . We therefore have that subsets of the form (4) generate a proper filter of . By the ultrafilter principle, we may extend this filter to an ultrafilter .
By construction, we have
but in fact these inclusions are equalities since the left sides and right sides are ultrafilters. Put differently, we have established
in other words belongs to , i.e., , as was to be shown.
This completes the proof of the Main Theorem (theorem 1).
A function between two topological spaces is continuous if and only if for their respective topological notions of convergence .
Suppose first that is continuous, and that belongs to , i.e., there is such that and . We want to show , or that any open set containing belongs to . The latter means , which is true since is an open set containing and .
Now suppose . To show is continuous, it suffices to show that
for any (easy exercise). For , lemma 1 shows there is with and . Under the supposition we have , and we also have , because and is upward closed and implies . Then again by lemma 1, and implies , as desired.
The category of topological spaces is equivalent (even isomorphic to) the category of lax -modules and lax morphisms between them.
A relational -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 ) precisely a relational -module in which every ultrafilter converges to exactly one point, that is in which the action of the monad lives in rather than in . Full proofs may be found at compactum; see also ultrafilter monad.
A continuous map from to is proper if the square
commutes (strictly) in , and is open if the square
commutes in . From this point of view, a space is Hausdorff if the diagonal map is proper, and compact if is proper (and these facts remain true even for pseudotopological spaces). See Clementino, Hofmann, and Janelidze, infra corollary 2.5.
The following ultrafilter interpolation result is due to Pisani:
A topological space is exponentiable if, whenever for and , there exists with and .
For an convergence-approach extension of this result to exponentiable maps in , see Clementino, Hofmann, and Tholen.
A continuous map is a discrete fibration if, whenever in and , there exists a unique such that and in .
A continuous map is étale (a local homeomorphism) if and are both discrete fibrations.
For more on this, see Clementino, Hofmann, and Janelidze.
In nonstandard analysis (which implicitly relies throughout on ), one may define a topological space using a relation between hyperpoints (elements of ) and standard points (elements of ). If is a hyperpoint and is a standard point, then we write and say that is a standard part? of or that belongs to the halo? (or monad, but not the category-theoretic kind) of . This relation must satisfy a condition analogous to the condition in the definition of a relational -module. The nonstandard defintions of open set, compact space, etc are also analogous. (Accordingly, one can speak of the standard part of 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.
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 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.
For more on relations between Barr’s approach to topological spaces, Lawvere’s approach to metric spaces, as well as uniform structures, prometric spaces, and approach structures?, see Clementino, Hofmann, and Tholen.