relational beta-module

Relational β\beta-modules


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 β\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 β\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 (UPUP). However, the definition can be treated on its own, even in a context without UPUP. So we also consider the properties of relational β\beta-modules when these might not match Bourbaki spaces.


Abstract description

If SS is a set, let βS\beta{S} be the set of ultrafilters on SS. This set is canonically identified with the set of Boolean algebra homomorphisms

P(S)2,P(S) \to \mathbf{2},

from the power set of SS to 2\mathbf{2}, the unique Boolean algebra with two elements. The 2-element set carries a dualizing object structure that induces an evident adjoint pair

(SetPBool op)(Bool ophom(,2)Set)(Set \stackrel{P}{\to} Bool^{op}) \; \dashv \; (Bool^{op} \stackrel{\hom(-, \mathbf{2})}{\to} Set)

so that the composite functor β=hom(P,2):SetSet\beta = \hom(P-, \mathbf{2}): Set \to Set carries a monad structure.

The functor β:SetSet\beta : Set \to Set extends to Rel as follows: given a binary relation r:XYr\colon X \to Y, written as a subobject in SetSet

Rπ 1,π 2X×Y,R \stackrel{\langle \pi_1, \pi_2 \rangle}{\to} X \times Y,

we define β(r):β(X)β(Y)\beta(r): \beta(X) \to \beta(Y) to be the relation obtained by taking the image of β(π 1),β(π 2):β(R)β(X)×β(Y)\langle \beta(\pi_1), \beta(\pi_2) \rangle: \beta(R) \to \beta(X) \times \beta(Y). It turns out, although it is by no means obvious, that β\beta is according to this definition a strict functor on RelRel.

The monad structure on β:SetSet\beta: Set \to Set, given by a unit u:1βu: 1 \to \beta and multiplication m:βββm: \beta \beta \to \beta, extends not to a strict monad on RelRel, but rather one where the transformations u,mu, m are op-lax in the sense of there being inequalities

X u X βX ββX m X βX r β(r) ββ(r) β(r) Y u Y βY ββY m Y βY\array{ X & \stackrel{u_X}{\to} & \beta X & & & & & & \beta \beta X & \stackrel{m_X}{\to} & \beta X \\ \mathllap{r} \downarrow & \leq & \downarrow \mathrlap{\beta(r)} & & & & & & \mathllap{\beta \beta (r)} \downarrow & \leq & \downarrow \mathrlap{\beta(r)} \\ Y & \underset{u_Y}{\to} & \beta Y & & & & & & \beta \beta Y & \underset{m_Y}{\to} & \beta Y }

(while of course the monad associativity and unit conditions remain as equations: hold on the nose).

Then a relational β\beta-module is a lax algebra? (module) of β\beta on the 2-poset RelRel. In other words, a set SS equipped with a relation ξ:βSS\xi: \beta S \to S such that the following inequalities hold:

(1)S u S βS ββS m S βS 1 S ξ β(ξ) ξ S βS ξ S \array{ S & \stackrel{u_S}{\to} & \beta S & & & & & & \beta \beta S & \stackrel{m_S}{\to} & \beta S \\ & \mathllap{1_S} \searrow \; \leq & \downarrow \mathrlap{\xi} & & & & & & \mathllap{\beta(\xi)} \downarrow & \leq & \downarrow \mathrlap{\xi} \\ & & S & & & & & & \beta S & \underset{\xi}{\to} & S }

Arguably, it is better to consider RelRel 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 β\beta-modules. We touch on this below, but for a much wider context, see generalized multicategory.

Bridge to a concrete description

A relational β\beta-module is a set SS and a binary relation ξ:βSS\xi: \beta S \to S between ultrafilters on SS and elements of SS that satisfy the conditions (1) . For FβSF \in \beta S and xSx \in S, we write F ξxF \rightsquigarrow_\xi x if (F,x)(F, x) satisfies the relation ξ\xi, or often just FxF \rightsquigarrow x if the relation is clear. We pronounce this by saying “the ultrafilter FF converges to the point xx”, so that ξ\xi plays the role of “notion of convergence”.

Preliminary to explaining the conditions (1), we first set up a Galois connection between ξRel(βS,S)\xi \in Rel(\beta S, S) and subsets 𝒞PP(S)\mathcal{C} \in P P(S), so that fixed points on the PP(S)P P(S) side are exactly topologies on SS, and fixed points on the other side are (as we show below) lax β\beta-module structures on SS. The Galois connection would then of course restrict to a Galois correspondence between topologies and lax module structures.

Recall that each topology 𝒪P(S)\mathcal{O} \subseteq P(S) induces a notion of convergence where FxF \rightsquigarrow x means N xFN_x \subseteq F (FF contains the filter of neighborhoods of xx). Accordingly, for general 𝒞P(S)\mathcal{C} \subseteq P(S), define the relation conv(𝒞)=ξ:βSSconv(\mathcal{C}) = \xi: \beta S \to S by

F ξx( U:P(S))U𝒞xUUF.F \rightsquigarrow_\xi x \;\;\; \Leftrightarrow \;\;\; (\forall_{U: P(S)})\; U \in \mathcal{C} \; \wedge \; x \in U \; \Rightarrow \; U \in F.

Conversely, a topology 𝒪\mathcal{O} can be retrieved from its notion of convergence: under the ultrafilter principle, the neighborhood filter of a point xx is just the intersection of all ultrafilters containing it (hence all FF such that FxF \rightsquigarrow x), and then a set is open if it is a neighborhood of all of its elements. Accordingly, for general “notions of convergence” ξRel(βS,S)\xi \in Rel(\beta S, S), we define a collection τ(ξ)P(S)\tau(\xi) \subseteq P(S) by

τ(ξ){US:( F:βS,x:S)(xUF ξx)UF}.\tau(\xi) \coloneqq \{U \subseteq S: \; (\forall_{F: \beta S, x: S})\; (x \in U \; \wedge\; F \rightsquigarrow_\xi x) \Rightarrow U \in F\}.

τ(ξ)\tau(\xi) is a topology on SS, for any ξ:Rel(βS,S)\xi: Rel(\beta S, S).


It is trivial that Sτ(ξ)S \in \tau(\xi). If U,Vτ(ξ)U, V \in \tau(\xi), and if xUVx \in U \cap V and FxF \rightsquigarrow x, then also xUx \in U and xVx \in V and we conclude U,VFU, V \in F, whence UVFU \cap V \in F since FF is an ultrafilter, so that UVU \cap V satisfies the condition of belonging to τ(ξ)\tau(\xi). Given a collection of elements U iτ(ξ)U_i \in \tau(\xi), if x iU ix \in \cup_i U_i and FxF \rightsquigarrow x, then xU ix \in U_i for some ii and we conclude U iFU_i \in F, whence iU iF\cup_i U_i \in F since FF is upward closed. Therefore iU i\cup_i U_i satisfies the condition of belonging to τ(ξ)\tau(\xi) (vacuously so if the collection is empty).


There is a Galois connection between notions of convergence on SS and subsets of P(S)P(S), according to the bi-implication

𝒞τ(ξ)ξconv(𝒞).\mathcal{C} \subseteq \tau(\xi) \; \Leftrightarrow \; \xi \subseteq conv(\mathcal{C}).

To establish the bi-implication, it suffices to observe that both containments 𝒞τ(ξ)\mathcal{C} \subseteq \tau(\xi) and ξconv(𝒞)\xi \subseteq conv(\mathcal{C}) are equivalent to the condition

F:βS U:PS x:S(F ξx)(U𝒞)(xU)(UF).\forall_{F: \beta S} \forall_{U: P S} \forall_{x: S} (F \rightsquigarrow_\xi x) \; \wedge \; (U \in \mathcal{C}) \; \wedge \; (x \in U) \; \; \Rightarrow \; \; (U \in F).

If 𝒪\mathcal{O} is a topology on SS, then 𝒪=τ(conv(𝒪))\mathcal{O} = \tau(conv(\mathcal{O})) (i.e., topologies are fixed points of the closure operator τconv\tau \circ \conv).


We already have 𝒪τ(conv(𝒪))\mathcal{O} \subseteq \tau(conv(\mathcal{O})) from Proposition 2. For the other direction, we must show that any VV belonging to τ(conv(𝒪))\tau(conv(\mathcal{O})) is an 𝒪\mathcal{O}-neighborhood of each of its points. Suppose the contrary: that xVx \in V but VV is not an 𝒪\mathcal{O}-neighborhood of xx. Then for every 𝒪\mathcal{O}-neighborhood UN xU \in N_x, we have U¬VU \cap \neg V \neq \emptyset, so that sets of this form generate a filter. By the ultrafilter principle, we may extend this filter to an ultrafilter FF; clearly we have FxF \rightsquigarrow x and ¬VF\neg V \in F, but since FxF \rightsquigarrow x and Vτ(conv(𝒪))V \in \tau(conv(\mathcal{O})) and xVx \in V, we also have VFV \in F, which is inconsistent with ¬VF\neg V \in F.

Propositions 1, 2, and 3 more or less show that a topological space (S,𝒪)(S, \mathcal{O}) is a particular type of pseudotopological space:


A pseudotopological space is set SS equipped with a relation ξ:βSS\xi: \beta S \to S such that 1 Sξu S1_S \leq \xi \circ u_S.

All that remains is to check is:


The lax unit condition 1 Sξu S1_S \leq \xi \circ u_S holds if ξ=conv(𝒪)\xi = conv(\mathcal{O}), for a topology 𝒪\mathcal{O}.

The unit u S:SβSu_S: S \to \beta S may also be denoted prin Sprin_S, as it takes an element xSx \in S to the principal ultrafilter

prin S(x)={US:xU}prin_S(x) = \{U \subseteq S: x \in U\}

and now the unit condition says prin S(x) ξxprin_S(x) \rightsquigarrow_\xi x for all xx. For ξ=conv(𝒪)\xi = conv(\mathcal{O}), this says N xprin S(x)N_x \subseteq prin_S(x), or that xVx \in V for all neighborhoods VN xV \in N_x, which is a tautology.

One of our goals is to prove the following theorem:


(Main Theorem) An arrow ξ:β(S)S\xi: \beta(S) \to S in RelRel is of the form conv(τ(ξ))conv(\tau(\xi)) if and only if the following inequalities are satisfied:

1 Sξprin S,ξβ(ξ)ξm S1_S \leq \xi \circ prin_S, \qquad \xi \circ \beta(\xi) \leq \xi \circ m_S

where m S:ββ(S)β(S)m_S: \beta \beta(S) \to \beta(S) is the multiplication on the ultrafilter monad.

Ultrafilter monad on RelRel

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 β\beta-module, leading up to the context of generalized multicategories.

Extending the ultrafilter functor to RelRel

First we examine more closely the extension of the ultrafilter functor β:SetSet\beta: Set \to Set to RelRel, showing in particular that the extension is a strict functor. First we slightly rephrase our earlier definition:


For a relation r:XYr: X \to Y between sets, given by a subobject RX×YR \hookrightarrow X \times Y in SetSet with projections f:RXf: R \to X and g:RYg: R \to Y, define β¯(r)\bar{\beta}(r) to be the composite

β(X)β(f) oβ(R)β(g)β(Y)\beta(X) \stackrel{\beta(f)^{o}}{\to} \beta(R) \stackrel{\beta(g)}{\to} \beta(Y)

in the bicategory of relations.

Any span of functions (h:SX,k:SY)(h: S \to X, k: S \to Y) that represents rr (in the sense that r=kh or = k h^{o} in the bicategory of relations) would serve in place of (f,g)(f, g), since for any such span there is an epi s:SRs: S \to R with h=fsh = f s, k=gsk = g s, whence β(s)\beta(s) is epi (because the epi ss splits in SetSet) and we have

β(g)β(f) o = β(g)β(s)β(s) oβ(f) o = β(g)β(s)(β(f)β(s)) o = β(gs)β(fs) o = β(k)β(h) o.\array{ \beta(g)\beta(f)^{o} & = & \beta(g)\beta(s)\beta(s)^{o}\beta(f)^{o} \\ & = & \beta(g)\beta(s)(\beta(f)\beta(s))^{o} \\ & = & \beta(g s)\beta(f s)^{o} \\ & = & \beta(k)\beta(h)^{o}. }

In particular, β¯\bar{\beta} is well-defined. Since β¯\bar{\beta} extends β:SetSet\beta: Set \to Set, there is no harm in writing β(r)\beta(r) in place of β¯(r)\bar{\beta}(r). If rr:XYr \leq r': X \to Y, then β(r)β(r)\beta(r) \leq \beta(r') (as can be seen from the calculation displayed above, but replacing the epi ss by a general map tt, and the first equation by an inequality \geq).


The same recipe works to extend any functor T:SetSetT: Set \to Set to RelRel, and the extension T¯:RelRel\bar{T}: Rel \to Rel is always an op-lax functor in the sense that

T¯(rs)T¯(r)T¯(s)\bar{T}(r s) \leq \bar{T}(r) \bar{T}(s)

as is easily seen by contemplating a pullback diagram (where r=gf or = g f^{o} and s=kh os = k h^{o}):

Q p q R S f g h k X Y Z\array{ & & & & Q & & & & \\ & & & \mathllap{p} \swarrow & & \searrow \mathrlap{q} & & & \\ & & R & & & & S & & \\ & \mathllap{f} \swarrow & & \searrow \mathrlap{g} & & \mathllap{h} \swarrow & & \searrow \mathrlap{k} & \\ X & & & & Y & & & & Z }

whereupon one calculates

T¯(rs) = T(kq)T(fp) o = T(k)T(q)T(p) oT(f) o T(k)T(h) oT(g)T(f) o = T¯(r)T¯(s)\array{ \bar{T}(r s) & = & T(k q) T(f p)^{o} \\ & = & T(k) T(q) T(p)^{o} T(f)^{o} \\ & \leq & T(k) T(h)^{o} T(g) T(f)^{o} \\ & = & \bar{T}(r) \bar{T}(s) }

where the inequality comes from T(q)T(p) oT(h) oT(g)T(q) T(p)^{o} \leq T(h)^{o} T(g), which is equivalent to T(h)T(q)T(g)T(p)T(h) T(q) \leq T(g) T(p) (where even equality holds). This calculation shows that T¯\bar{T} is an actual (not just an op-lax) functor on RelRel iff TT satisfies the Beck-Chevalley condition: if (p,q)(p, q) is a pullback of (g,h)(g, h), then

T(q)T(p) o=T(h) oT(g).T(q) T(p)^{o} = T(h)^{o} T(g).

This in turn amounts to TT preserving weak pullbacks. (It actually says TT takes pullbacks to weak pullbacks, but this implies TT takes weak pullbacks to weak pullbacks because any endofunctor TT on SetSet preserves epis, using the axiom of choice.)


The functor β:SetSet\beta: Set \to Set satisfies the Beck-Chevalley condition (and therefore the extension β:RelRel\beta: Rel \to Rel is a strict functor).


Referring to the pullback diagram in Remark 1, let Q=R× YSQ = R \times_Y S be the pullback. We must show that the canonical map

β(R× YS)β(R)× β(Y)β(S)\beta(R \times_Y S) \to \beta(R) \times_{\beta(Y)} \beta(S)

is epic. Viewing this as a continuous map between compact Hausdorff spaces (see this section of the article on compacta), it suffices to show that the canonical map

R× YSβ(R)× β(Y)β(S)R \times_Y S \to \beta(R) \times_{\beta(Y)} \beta(S)

has a dense image. Let (G,H)β(R)× β(Y)β(S)(G, H) \in \beta(R) \times_{\beta(Y)} \beta(S), so that β(g)(G)=β(h)(H)\beta(g)(G) = \beta(h)(H) are the same ultrafilter Jβ(Y)J \in \beta(Y). Let A^\hat{A} and B^\hat{B} be basic open neighborhoods of GG and HH in β(R)\beta(R) and β(S)\beta(S) respectively; we must show that there is (r,s)R× YS(r, s) \in R \times_Y S such that

(prin(r),prin(s))A^×B^(prin(r), prin(s)) \in \hat{A} \times \hat{B}

or in other words such that rAr \in A and sBs \in B. We have g 1(g(A))Gg^{-1}(g(A)) \in G since AGA \in G and Ag 1(g(A))A \subseteq g^{-1}(g(A)), so that g(A)g(A) belongs to

J=β(g)(G){CY:g 1(C)G}J = \beta(g)(G) \coloneqq \{C \subseteq Y: g^{-1}(C) \in G\}

and similarly h(B)Jh(B) \in J. It follows that g(A)h(B)Jg(A) \cap h(B) \in J so that g(A)h(B)g(A) \cap h(B) \neq \emptyset. Any element yg(A)h(B)y \in g(A) \cap h(B) can be written as y=g(r)y = g(r) and y=h(s)y = h(s) for some rAr \in A and sBs \in B, and this completes the proof.

Ultrafilter monad on the equipment Rel\mathbf{Rel}

As mentioned in an earlier section, the natural transformations u=prin:1 Setβu = prin: 1_{Set} \to \beta, m:βββm: \beta\beta \to \beta do not extend to (strict) natural transformations on the locally posetal bicategory RelRel, but only to transformations that are op-lax in the sense of inequalities

prin Yrβ(r)prin X,m Yββ(r)β(r)m Xprin_Y \circ r \leq \beta(r) \circ prin_X, \qquad m_Y \circ \beta\beta(r) \leq \beta(r) \circ m_X

for every relation r:XYr: X \to Y. These are equivalent to inequalities

rprin Y oβ(r)prin X,ββ(r)m Y oβ(r)m Xr \leq prin_Y^o \circ \beta(r) \circ prin_X, \qquad \beta\beta(r) \leq m_Y^o \circ \beta(r) \circ m_X

and they may be deduced simply by staring at naturality diagrams in SetSet, in which we represent or tabulate rr by π 2π 1 o\pi_2 \circ \pi_1^o:

R ββ(R) π 1 prin R π 2 ββπ 1 m R ββπ 2 X β(R) Y ββ(X) β(R) ββ(Y) prin X βπ 1 βπ 2 prin Y m X βπ 1 βπ 2 m Y β(X) β(Y) β(X) β(Y)\array{ & & R & & & & & & & & & & \beta\beta (R) & & \\ & \mathllap{\pi_1} \swarrow & \downarrow_\mathrlap{prin_R} & \searrow \mathrlap{\pi_2} & & & & & & & & _\mathllap{\beta\beta\pi_1} \swarrow & \downarrow_\mathrlap{m_R} & \searrow_\mathrlap{\beta\beta\pi_2} & \\ X & & \beta (R) & & Y & & & & & & \beta \beta (X) & & \beta (R) & & \beta \beta (Y) \\ _\mathllap{prin_X} \downarrow & \swarrow_\mathrlap{\beta \pi_1} & & _\mathllap{\beta \pi_2} \searrow & \downarrow_\mathrlap{prin_Y} & & & & & & \mathllap{m_X} \downarrow & \swarrow_\mathrlap{\beta \pi_1} & & _\mathllap{\beta \pi_2} \searrow & \downarrow \mathrlap{m_Y} \\ \beta (X) & & & & \beta (Y) & & & & & & \beta (X) & & & & \beta (Y) }

To get an actual monad, it is more satisfactory in this context to consider not the bicategory RelRel, but rather the equipment or framed bicategory Rel\mathbf{Rel}. That is, there is a 2-category EquipEquip 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 β¯:RelRel\bar{\beta}: \mathbf{Rel} \to \mathbf{Rel}.

In more detail: the 0-cells of Rel\mathbf{Rel} are sets, and the horizontal arrows are relations between sets. Vertical arrows are functions between sets, and a 2-cell of shape

A r B f g C s D;\array{ A & \stackrel{r}{\to} & B \\ \mathllap{f} \downarrow & \Downarrow & \downarrow \mathrlap{g} \\ C & \stackrel{s}{\to} & D; }

is an inequality grsfg \circ r \leq s \circ f. We straightforwardly get a double category Rel\mathbf{Rel}, and the ultrafilter functor on SetSet extends to a functor β¯:RelRel\bar{\beta}: \mathbf{Rel} \to \mathbf{Rel} 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 F,G:BCF, G: \mathbf{B} \to \mathbf{C} between equipments. A transformation η:FG\eta: F \to G assigns to each 0-cell bb of B\mathbf{B} a vertical arrow ηb:FbGb\eta b: F b \to G b, and to each horizontal arrow r:bbr: b \to b' a 2-cell ηr\eta r of the form

Fb Fr Fb ηb ηr ηb Gb Gr Gb;\array{ F b & \stackrel{F r}{\to} & F b' \\ \mathllap{\eta b} \downarrow & \Downarrow \mathrlap{\eta r} & \downarrow \mathrlap{\eta b'} \\ G b & \stackrel{G r}{\to} & G b'; }

suitably compatible with the double category structures.

We thus find that the op-lax structures of the transformations prin:1β¯prin: 1 \to \bar{\beta}, m:β¯β¯β¯m: \bar{\beta} \bar{\beta} \to \bar{\beta} on RelRel qua bicategory are exactly what we need to produce honest transformations u:1β¯u: 1 \to \bar{\beta}, m:β¯β¯β¯m: \bar{\beta} \bar{\beta} \to \bar{\beta} on Rel\mathbf{Rel} qua equipment, and the result is an ultrafunctor monad on the equipment Rel\mathbf{Rel}.

Given a monad TT on an equipment B\mathbf{B}, one may proceed to construct a horizontal Kleisli equipment HKl(B,T)HKl(\mathbf{B}, T) with the same 0-cells and vertical arrows as B\mathbf{B}, but whose horizontal arrows are of the form r:bTbr: b \to T b'. A 2-cell in HKl(B,T)HKl(\mathbf{B}, T) (with vertical source ff and vertical target gg) is a 2-cell in B\mathbf{B} of the form

b r Tb f α Tg c s Tc;\array{ b & \stackrel{r}{\to} & T b' \\ \mathllap{f} \downarrow & \Downarrow \mathrlap{\alpha} & \downarrow \mathrlap{T g} \\ c & \stackrel{s}{\to} & T c'; }

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 m b:TTbTbm_b: T T b \to T b, 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 ξ:bb\xi: b \to b together with unit and multiplication 2-cells

b 1 b b b ξbξ b 1 η 1 1 μ 1 b ξ b b ξ b\array{ b & \stackrel{1_b}{\to} & b & & & & & & b & \stackrel{\xi}{\to}\;\;\; b \;\;\; \stackrel{\xi}{\to} & b\\ \mathllap{1} \downarrow & \Downarrow \mathrlap{\eta} & \downarrow \mathrlap{1} & & & & & & \mathllap{1} \downarrow & \Downarrow \mathrlap{\mu} & \downarrow \mathrlap{1}\\ b & \stackrel{\xi}{\to} & b & & & & & & b & \stackrel{\xi}{\to} & b }

satisfying evident identities. A monoid homomorphism from (b,ξ)(b, \xi) to (c,θ)(c, \theta) consists of a vertical arrow and 2-cell (f,ϕ)(f, \phi) of the form

b ξ b f ϕ f c θ c\array{ b & \stackrel{\xi}{\to} & b \\ \mathllap{f} \downarrow & \Downarrow \mathrlap{\phi} & \downarrow \mathrlap{f} \\ c & \underset{\theta}{\to} & c }

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 β\beta-modules.


Given a monad TT on an equipment B\mathbf{B}, a TT-monoid is a monoid in the horizontal Kleisli equipment HKl(B,T)HKl(\mathbf{B}, T). A map of TT-monoids is a homomorphism between monoids in HKl(B,T)HKl(\mathbf{B}, T).


For the ultrafilter monad β\beta on the equipment Rel\mathbf{Rel}, a structure of β\beta-monoid is equivalent to a structure of relational β\beta-module, and a homomorphism of β\beta-monoids is the same as a lax map of relational β\beta-modules in the bicategory RelRel.


This is really just a matter of unwinding definitions. The data of a β\beta-monoid in the equipment Rel\mathbf{Rel} amounts to a set XX together with a horizontal arrow in the Kleisli construction, that is to say a relation c:XβXc: X \to \beta X (opposite to our conventional direction, i.e., c=ξ oc = \xi^o). The unit and multiplication cells for cc are inequalities 1 Xc1_X \leq c and c Klccc \circ_{Kl} c \leq c (the vertical source and target being identity maps), where the identity 1 X1_X in the Kleisli construction uses the unit for β\beta and the Kleisli composition uses the multiplication. Back in the bicategory RelRel these translate to relational inequalities

prin Xc,m Xβcccprin_X \leq c, \qquad m_X \circ \beta c \circ c \leq c

or, with c=ξ oc = \xi^o,

prin Xξ o,m X(β(ξ)) oξ oξ o.prin_X \leq \xi^o, \qquad m_X \circ (\beta (\xi))^o \circ \xi^o \leq \xi^o.

These boil down to relational inequalities

1 Xprin X oξ o,(β(ξ)) oξ om X oξ o1_X \leq prin_X^o \circ \xi^o, \qquad (\beta (\xi))^o \circ \xi^o \leq m_X^o \circ \xi^o

or to

1 Xξprin X,ξβ(ξ)ξm X,1_X \leq \xi \circ prin_X, \qquad \xi \circ \beta (\xi) \leq \xi \circ m_X,

as in the axioms on relational beta-modules. Similarly, a β\beta-monoid homomorphism (X,c)(Y,d)(X, c) \to (Y, d) is a vertical arrow f:XYf: X \to Y in HKl(Rel,β)HKl(\mathbf{Rel}, \beta) together with a suitable 2-cell, which after some unraveling comes down to a relational inequality

β(f)cdf\beta (f) \circ c \leq d \circ f

or to an inequality β(f)ξ X oξ Y of\beta (f) \circ \xi_X^o \leq \xi_Y^o \circ f, which may be further massaged into the form ξ X of o(β(f)) oξ Y o\xi_X^o \circ f^o \leq (\beta (f))^o \circ \xi_Y^o, or simply to

fξ Xξ Yβ(f)f \circ \xi_X \leq \xi_Y \circ \beta (f)

as advertised in the notion of lax morphism of relational β\beta-modules (cf. theorem 4 below).


In some sense, relational β\beta-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 TT acting on virtual equipments and study normalized TT-monoids. Explaining all this requires a lengthy build-up. Even in the relatively restricted packet of unifications that come under the rubric of (T,V)(T, V)-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 VV-matrices) can be somewhat elaborate, and mildly daunting.

The example of the ultrafilter monad acting on RelRel 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.

Proof of Main Theorem

We now return to the task of proving theorem 1.


For a topological space (S,𝒪)(S, \mathcal{O}) and a point xSx \in S, let ξ=conv(𝒪)\xi = conv(\mathcal{O}), and let 𝒪 x\mathcal{O}_x be the collection of open neighborhoods of xx. Then F ξxF \rightsquigarrow_\xi x, i.e., N xFN_x \subseteq F, is equivalent to 𝒪 xF\mathcal{O}_x \subseteq F. This is because N xN_x is the filter generated by 𝒪 x\mathcal{O}_x.


The following conditions are equivalent:

  • ξ=conv(𝒞)\xi = conv(\mathcal{C}) for some 𝒞P(S)\mathcal{C} \subseteq P(S);

  • ξ=conv(𝒪)\xi = conv(\mathcal{O}) for some topology 𝒪P(S)\mathcal{O} \subseteq P(S);

  • ξ=conv(τ(ξ))\xi = conv(\tau(\xi)).

Indeed, convτconv=convconv \circ \tau \circ conv = conv by general properties of Galois connections. Applying convτconv \circ \tau to both sides of the first equation, we have

conv(τ(ξ))=(convτconv)(𝒞)=conv(𝒞)=ξconv(\tau(\xi)) = (conv \circ \tau \circ conv)(\mathcal{C}) = conv(\mathcal{C}) = \xi

so the first equation implies the third. Which in turn implies the second, since we know by proposition 1 that collections 𝒪\mathcal{O} of the form τ(ξ)\tau(\xi) are topologies. The second equation trivially implies the first.

We now break up our Main Theorem 1 into the following two theorems.


If ξ=conv(𝒪)\xi = conv(\mathcal{O}) for a topology 𝒪\mathcal{O}, 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 ξ\xi by a span βSπ 1Rπ 2S\beta S \stackrel{\pi_1}{\leftarrow} R \stackrel{\pi_2}{\to} S, so that β(ξ)=β(π 2)β(π 1) o\beta(\xi) = \beta(\pi_2) \beta(\pi_1)^o. The lax associativity condition becomes

π 2π 1 oβ(π 2)β(π 1) oπ 2π 1 om S\pi_2 \pi_1^o \beta(\pi_2) \beta(\pi_1)^o \leq \pi_2 \pi_1^o m_S

which (using β(π 1)β(π 1) o\beta(\pi_1) \dashv \beta(\pi_1)^o) is equivalent to

(2)π 2π 1 oβ(π 2)π 2π 1 om Sβ(π 1) \pi_2 \pi_1^o \beta(\pi_2) \leq \pi_2 \pi_1^o m_S \beta(\pi_1)

or in other words that for all 𝒢:β(R)\mathcal{G}: \beta(R), x:Sx: S

(3)β(π 2)(𝒢) ξxm Sβ(π 1)(𝒢) ξx. \beta(\pi_2)(\mathcal{G}) \rightsquigarrow_\xi x \;\; \vdash \;\; m_S \beta(\pi_1)(\mathcal{G}) \rightsquigarrow_\xi x.

Here β(π 2)(𝒢)\beta(\pi_2)(\mathcal{G}) is, by definition,

{US:π 2 1(U)𝒢},\{U \subseteq S: \pi_2^{-1}(U) \in \mathcal{G}\},

with β(π 1)(𝒢)\beta(\pi_1)(\mathcal{G}) defined similarly. The monad multiplication m S:ββSβSm_S: \beta \beta S \to \beta S is by definition

(𝒰:ββS)m S(𝒰){AS:A^𝒰}(\mathcal{U}: \beta\beta S) \;\; m_S(\mathcal{U}) \coloneqq \{A \subseteq S: \hat{A} \in \mathcal{U}\}

where A^={FβS:AF}\hat{A} = \{F \in \beta S: A \in F\} (see also the previous section).

Thus, (3) translates into the following entailment (using remark 3):

𝒪 x{AS:π 2 1(A)𝒢} U:PSU𝒪 xπ 1 1(U^)𝒢.\array{ & & \mathcal{O}_x \subseteq \{A \subseteq S: \pi_2^{-1}(A) \in \mathcal{G}\} \\ & \vdash & \forall_{U: P S} U \in \mathcal{O}_x \Rightarrow \pi_1^{-1}(\hat{U}) \in \mathcal{G}. }

This would naturally follow if

U𝒪 xπ 2 1(U)π 1 1(U^).\forall_{U \in \mathcal{O}_x} \pi_2^{-1}(U) \subseteq \pi_1^{-1}(\hat{U}).

But a pair (F,y)(F, y) belongs to π 2 1(U)\pi_2^{-1}(U) if F ξyF \rightsquigarrow_\xi y and yUy \in U; we want to show this implies F=π 1(F,y)F = \pi_1(F, y) belongs to U^\hat{U}, or in other words that UFU \in F. But this is tautological, given how conv(𝒪)conv(\mathcal{O}) is defined in terms of a topology 𝒪\mathcal{O}.

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 ξ:Rel(βS,S)\xi: Rel(\beta S, S) and x:Sx: S, ASA \subseteq S, we have that xx belongs to the closure A¯\bar{A} wrt the topology τ(ξ)\tau(\xi) if and only if F:βSAFF ξx\exists_{F: \beta S} A \in F \; \wedge \; F \rightsquigarrow_\xi x.


As usual, let ¬A\neg A denote the complement of a subset AA. By definition of the topology τ(ξ)\tau(\xi), we have that ¬A\neg A is a neighborhood of xx if F:βSF ξx¬AF\forall_{F: \beta S} F \rightsquigarrow_\xi x \; \Rightarrow \; \neg A \in F. In other words,

xint(¬A) F:βSF ξx¬AF F:βS¬((AF)(F ξx))\array{ x \in int(\neg A) & \Leftrightarrow & \forall_{F: \beta S} \; F \rightsquigarrow_\xi x \; \Rightarrow \; \neg A \in F \\ & \Leftrightarrow & \forall_{F: \beta S} \; \neg ((A \in F) \; \wedge \; (F \rightsquigarrow_\xi x)) }

since in an ultrafilter FF, we have ¬(AF)\neg (A \in F) iff (¬A)F(\neg A) \in F. Negating both sides of this bi-implication gives

xA¯ F:βSAFF ξx\array{ x \in \bar{A} & \Leftrightarrow & \exists_{F: \beta S} \; A \in F \; \wedge \; F \rightsquigarrow_\xi x }

as desired.


If ξ:βSS\xi: \beta S \to S in RelRel satisfies the inequalities of (1), then ξ=conv(τ(ξ))\xi = conv(\tau(\xi)).


We have ξconv(τ(ξ))\xi \leq conv(\tau(\xi)) from the Galois connection (proposition 2), so we just need to prove conv(τ(ξ))ξconv(\tau(\xi)) \leq \xi, or that F conv(τ(ξ))xF \rightsquigarrow_{conv(\tau(\xi))} x (henceforth abbreviated as F τ(ξ)xF \rightsquigarrow_{\tau(\xi)} x) implies F ξxF \rightsquigarrow_\xi x under the conditions (1).

If F τ(ξ)xF \rightsquigarrow_{\tau(\xi)} x, then every neighborhood VV of xx belongs to FF, so that for every UFU \in F, every neighborhood VV of xx intersects UU in a nonempty set. But this just means xU¯x \in \bar{U} for every UFU \in F, or in other words (using lemma 1) that

UF G:βSUGG ξx.U \in F \; \; \vdash \; \; \exists_{G: \beta S} U \in G \; \wedge \; G \rightsquigarrow_\xi x.

Representing the relation ξ\xi as usual by a subset π 1,π 2:RβS×S\langle \pi_1, \pi_2 \rangle: R \hookrightarrow \beta S \times S, another way of expressing the existential formula on the right of this entailment is:

(G,x):βS×S(G,x)RGU^\exists_{(G, x): \beta S \times S} \; (G, x) \in R \; \wedge \; G \in \hat{U}


γ:Rπ 1(γ)U^π 2(γ)=x\exists_{\gamma: R} \pi_1(\gamma) \in \hat{U} \wedge \pi_2(\gamma) = x

or even just

(4)π 1 1(U^)π 2 1({x}) \pi_1^{-1}(\hat{U}) \wedge \pi_2^{-1}(\{x\}) \neq \emptyset

as subsets of RR, as UU ranges over all elements of FF. We therefore have that subsets of the form (4) generate a proper filter of RR. By the ultrafilter principle, we may extend this filter to an ultrafilter 𝒢βR\mathcal{G} \in \beta R.

By construction, we have

F{BX:π 1 1(B^))𝒢}prin S(x){AX:π 2 1(A)𝒢}F \subseteq \{B \subseteq X: \pi_1^{-1}(\hat{B})) \in \mathcal{G}\} \qquad prin_S(x) \subseteq \{A \subseteq X: \pi_2^{-1}(A) \in \mathcal{G}\}

but in fact these inclusions are equalities since the left sides and right sides are ultrafilters. Put differently, we have established

F=(m Sβ(π 1))(𝒢),prin S(x)=β(π 2)(𝒢).F = (m_S \circ \beta(\pi_1))(\mathcal{G}), \qquad prin_S(x) = \beta(\pi_2)(\mathcal{G}).

Notice that the lax unit condition of (1) implies that prin S(x)=β(π 2)(𝒢) ξxprin_S(x) = \beta(\pi_2)(\mathcal{G}) \rightsquigarrow_\xi x, or that (𝒢,x)(\mathcal{G}, x) belongs to the relation ξβ(π 2)\xi \circ \beta(\pi_2). Recall also that the lax associativity condition is equivalent to (2), which says

ξβ(π 2)ξm Sβ(π 1);\xi \circ \beta(\pi_2) \leq \xi \circ m_S \circ \beta(\pi_1);

in other words (𝒢,x)(\mathcal{G}, x) belongs to ξm Sβ(π 1)\xi \circ m_S \circ \beta(\pi_1), i.e., F=(m Sβ(π 1))(𝒢) ξxF = (m_S \circ \beta(\pi_1))(\mathcal{G}) \rightsquigarrow_\xi x, as was to be shown.

This completes the proof of the Main Theorem (theorem 1).

Continuous maps


A function between two topological spaces f:XYf: X \to Y is continuous if and only if fξθβ(f)f \circ \xi \leq \theta \circ \beta(f) for their respective topological notions of convergence ξ,θ\xi, \theta.


Suppose first that ff is continuous, and that (F,y)β(X)×Y(F, y) \in \beta(X) \times Y belongs to fξf \circ \xi, i.e., there is xx such that FxF \rightsquigarrow x and f(x)=yf(x) = y. We want to show β(f)(F)y=f(x)\beta(f)(F) \rightsquigarrow y = f(x), or that any open set VV containing f(x)f(x) belongs to β(f)(F)\beta(f)(F). The latter means f 1(V)Ff^{-1}(V) \in F, which is true since f 1(V)f^{-1}(V) is an open set containing xx and FxF \rightsquigarrow x.

Now suppose fξθβ(f)f \circ \xi \leq \theta \circ \beta(f). To show ff is continuous, it suffices to show that

f(A¯)f(A)¯f(\bar{A}) \subseteq \widebar{f(A)}

for any AXA \subseteq X (easy exercise). For xA¯x \in \bar{A}, lemma 1 shows there is F:β(X)F: \beta(X) with AFA \in F and FxF \rightsquigarrow x. Under the supposition we have β(f)(F)f(x)\beta(f)(F) \rightsquigarrow f(x), and we also have f(A)β(f)(F)f(A) \in \beta(f)(F), because Af 1(f(A))A \subseteq f^{-1}(f(A)) and FF is upward closed and AFA \in F implies f 1(f(A))Ff^{-1}(f(A)) \in F. Then again by lemma 1, f(A)β(f)(F)f(A) \in \beta(f)(F) and β(f)(F)f(x)\beta(f)(F) \rightsquigarrow f(x) implies f(x)f(A)¯f(x) \in \widebar{f(A)}, as desired.

Corollary (Barr)

The category of topological spaces is equivalent (even isomorphic to) the category of lax β\beta-modules and lax morphisms between them.


As above, a subset AA of SS is open if A𝒰A \in \mathcal{U} whenever 𝒰xA\mathcal{U} \to x \in A. On the other hand, by lemma 1, AA is closed if xAx \in A whenever A𝒰xA \in \mathcal{U} \to x.

A relational β\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 UFUF) precisely a relational β\beta-module in which every ultrafilter converges to exactly one point, that is in which the action of the monad β\beta lives in SetSet rather than in RelRel. Full proofs may be found at compactum; see also ultrafilter monad.

A continuous map ff from (X,ξ:βXX)(X, \xi: \beta X \to X) to (Y,θ:βYY)(Y, \theta: \beta Y \to Y) is proper if the square

βX ξ X β(f) f βY θ Y\array{ \beta X & \stackrel{\xi}{\to} & X \\ \mathllap{\beta (f)} \downarrow & & \downarrow \mathrlap{f} \\ \beta Y & \underset{\theta}{\to} & Y }

commutes (strictly) in RelRel, and ff is open if the square

βX ξ X β(f) o f o βY θ Y\array{ \beta X & \stackrel{\xi}{\to} & X \\ \mathllap{\beta (f)^o} \uparrow & & \uparrow \mathrlap{f^o} \\ \beta Y & \underset{\theta}{\to} & Y }

commutes in RelRel. From this point of view, a space XX is Hausdorff if the diagonal map δ:XX×X\delta: X \to X \times X is proper, and compact if ϵ:X1\epsilon: X \to 1 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 (X,ξ)(X, \xi) is exponentiable if, whenever m X(𝒰) ξxm_X(\mathcal{U}) \rightsquigarrow_\xi x for 𝒰ββX\mathcal{U} \in \beta\beta X and xXx \in X, there exists FβXF \in \beta X with 𝒰 β(ξ)F\mathcal{U} \rightsquigarrow_{\beta(\xi)} F and F ξxF \rightsquigarrow_\xi x.

For an convergence-approach extension of this result to exponentiable maps in TopTop, see Clementino, Hofmann, and Tholen.


A continuous map f:XYf: X \to Y is a discrete fibration if, whenever GyG \rightsquigarrow y in YY and f(x)=yf(x) = y, there exists a unique Fβ(X)F \in \beta(X) such that β(f)(F)=G\beta(f)(F) = G and FxF \rightsquigarrow x in XX.


A continuous map f:XYf: X \to Y is étale (a local homeomorphism) if ff and δ f:XX× YX\delta_f: X \to X \times_Y X are both discrete fibrations.

For more on this, see Clementino, Hofmann, and Janelidze.

Relation to nonstandard analysis

In nonstandard analysis (which implicitly relies throughout on UFUF), one may define a topological space using a relation between hyperpoints (elements of S *S^*) and standard points (elements of SS). If uu is a hyperpoint and xx is a standard point, then we write uxu \approx x and say that xx is a standard part? of uu or that uu belongs to the halo? (or monad, but not the category-theoretic kind) of xx. This relation must satisfy a condition analogous to the condition in the definition of a relational β\beta-module. The nonstandard defintions of open set, compact space, etc are also analogous. (Accordingly, one can speak of the standard part of uu 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 β\beta is treated as a monad on SetSet instead of on RelRel, then its algebras are the compacta (the compact Hausdorff spaces), again assuming UPUP; see ultrafilter monad, and more especially compactum.

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.

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.


  • Michael Barr, Relational algebras, Springer Lecture Notes in Math. 137 (1970), 39-55.
  • Maria Manuel Clementino, Dirk Hofmann, and George Janelidze, On exponentiability of étale algebraic homomorphisms, Preprint 11-35, University of Coimbra. (pdf)
  • Maria Manuel Clementino, Dirk Hofmann, and Walter Tholen, The convergence approach to exponentiable maps, Portugaliae Mathematica (Nova Series), Vol. 60 Issue 2 (2003), 139-160. (web) (pdf)
  • Maria Manuel Clementino, Dirk Hofmann, and Walter Tholen, One Setting for All: Metric, Topology, Uniformity, Approach Structure. (pdf)
  • Gavin J. Seal, Canonical and op-canonical lax algebras, Theory and Applications of Categories, 14 (2005), 221–243. (web)
  • Claudio Pisani, Convergence in exponentiable spaces, Theory Appl. Categories 5 (1999), 148-162. (web)

Revised on August 10, 2013 07:59:45 by Toby Bartels (