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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{relational beta-module} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topology}{}\paragraph*{{Topology}}\label{topology} [[!include topology - contents]] \hypertarget{relational_modules}{}\section*{{Relational $\beta$-modules}}\label{relational_modules} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak \noindent\hyperlink{abstract}{Abstract description}\dotfill \pageref*{abstract} \linebreak \noindent\hyperlink{bridge_to_a_concrete_description}{Bridge to a concrete description}\dotfill \pageref*{bridge_to_a_concrete_description} \linebreak \noindent\hyperlink{functor}{Ultrafilter monad on $Rel$}\dotfill \pageref*{functor} \linebreak \noindent\hyperlink{extending_the_ultrafilter_functor_to_}{Extending the ultrafilter functor to $Rel$}\dotfill \pageref*{extending_the_ultrafilter_functor_to_} \linebreak \noindent\hyperlink{ultrafilter_monad_on_the_equipment_}{Ultrafilter monad on the equipment $\mathbf{Rel}$}\dotfill \pageref*{ultrafilter_monad_on_the_equipment_} \linebreak \noindent\hyperlink{proof_of_main_theorem}{Proof of Main Theorem}\dotfill \pageref*{proof_of_main_theorem} \linebreak \noindent\hyperlink{continuous_maps}{Continuous maps}\dotfill \pageref*{continuous_maps} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{relation_to_nonstandard_analysis}{Relation to nonstandard analysis}\dotfill \pageref*{relation_to_nonstandard_analysis} \linebreak \noindent\hyperlink{relation_to_other_topological_concepts}{Relation to other topological concepts}\dotfill \pageref*{relation_to_other_topological_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} \begin{quote}% 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 system|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 \emph{[[An Australian conspectus of higher categories]]} \end{quote} In \hyperlink{Barr}{1970}, [[Michael Barr]] gave an abstract definition of [[topological space]] based on a notion of [[convergence]] of [[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]] ($UP$). However, the definition can be treated on its own, even in a context without $UP$. So we also consider the properties of relational $\beta$-modules when these might not match [[topological space|Bourbaki spaces]]. \hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions} \hypertarget{abstract}{}\subsubsection*{{Abstract description}}\label{abstract} If $S$ is a [[set]], let $\beta{S}$ be the set of [[ultrafilters]] on $S$. This set is canonically identified with the set of Boolean algebra homomorphisms \begin{displaymath} P(S) \to \mathbf{2}, \end{displaymath} from the power set of $S$ to $\mathbf{2}$, the unique Boolean algebra with two elements. The 2-element set carries a [[dualizing object]] structure that induces an evident [[adjoint pair]] \begin{displaymath} (Set \stackrel{P}{\to} Bool^{op}) \; \dashv \; (Bool^{op} \stackrel{\hom(-, \mathbf{2})}{\to} Set) \end{displaymath} so that the composite [[functor]] $\beta = \hom(P-, \mathbf{2}): Set \to Set$ carries a [[monad]] structure. The functor $\beta : Set \to Set$ extends to [[Rel]] as follows: given a [[binary relation]] $r\colon X \to Y$, written as a subobject in $Set$ \begin{displaymath} R \stackrel{\langle \pi_1, \pi_2 \rangle}{\to} X \times Y, \end{displaymath} we define $\beta(r): \beta(X) \to \beta(Y)$ to be the relation obtained by taking the image of $\langle \beta(\pi_1), \beta(\pi_2) \rangle: \beta(R) \to \beta(X) \times \beta(Y)$. It turns out, although it is by no means obvious, that $\beta$ is according to this definition a \emph{strict} functor on $Rel$. The monad structure on $\beta: Set \to Set$, given by a unit $u: 1 \to \beta$ and multiplication $m: \beta \beta \to \beta$, extends not to a strict monad on $Rel$, but rather one where the transformations $u, m$ are [[lax natural transformation|op-lax]] in the sense of there being inequalities \begin{displaymath} \itexarray{ 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 } \end{displaymath} (while of course the monad associativity and unit conditions remain as equations: hold on the nose). Then a \textbf{relational $\beta$-module} is a [[lax algebra]] (module) of $\beta$ on the [[2-poset]] $Rel$. In other words, a set $S$ equipped with a relation $\xi: \beta S \to S$ such that the following inequalities hold: \begin{equation} \itexarray{ 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 } \label{rel1}\end{equation} Arguably, it is better to consider $Rel$ 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 $\beta$-modules. We touch on this below, but for a much wider context, see [[generalized multicategory]]. \hypertarget{bridge_to_a_concrete_description}{}\subsubsection*{{Bridge to a concrete description}}\label{bridge_to_a_concrete_description} A \textbf{relational $\beta$-module} is a [[set]] $S$ and a [[binary relation]] $\xi: \beta S \to S$ between [[ultrafilters]] on $S$ and [[elements]] of $S$ that satisfy the conditions \eqref{rel1} . For $F \in \beta S$ and $x \in S$, we write $F \rightsquigarrow_\xi x$ if $(F, x)$ satisfies the relation $\xi$, or often just $F \rightsquigarrow x$ if the relation is clear. We pronounce this by saying ``the ultrafilter $F$ \emph{converges} to the point $x$'', so that $\xi$ plays the role of ``notion of convergence''. Preliminary to explaining the conditions \eqref{rel1}, we first set up a [[Galois connection]] between $\xi \in Rel(\beta S, S)$ and subsets $\mathcal{C} \in P P(S)$, so that fixed points on the $P P(S)$ side are exactly topologies on $S$, and fixed points on the other side are (as we show below) lax $\beta$-module structures on $S$. The Galois connection would then of course restrict to a Galois correspondence between topologies and lax module structures. Recall that each topology $\mathcal{O} \subseteq P(S)$ induces a notion of convergence where $F \rightsquigarrow x$ means $N_x \subseteq F$ ($F$ contains the filter of neighborhoods of $x$). Accordingly, for general $\mathcal{C} \subseteq P(S)$, define the relation $conv(\mathcal{C}) = \xi: \beta S \to S$ by \begin{displaymath} F \rightsquigarrow_\xi x \;\;\; \Leftrightarrow \;\;\; (\forall_{U: P(S)})\; U \in \mathcal{C} \; \wedge \; x \in U \; \Rightarrow \; U \in F. \end{displaymath} Conversely, a topology $\mathcal{O}$ can be retrieved from its notion of convergence: under the ultrafilter principle, the neighborhood filter of a point $x$ is just the intersection of all ultrafilters containing it (hence all $F$ such that $F \rightsquigarrow x$), and then a set is open if it is a neighborhood of all of its elements. Accordingly, for general ``notions of convergence'' $\xi \in Rel(\beta S, S)$, we define a collection $\tau(\xi) \subseteq P(S)$ by \begin{displaymath} \tau(\xi) \coloneqq \{U \subseteq S: \; (\forall_{F: \beta S, x: S})\; (x \in U \; \wedge\; F \rightsquigarrow_\xi x) \Rightarrow U \in F\}. \end{displaymath} \begin{prop} \label{topology}\hypertarget{topology}{} $\tau(\xi)$ is a topology on $S$, for any $\xi: Rel(\beta S, S)$. \end{prop} \begin{proof} It is trivial that $S \in \tau(\xi)$. If $U, V \in \tau(\xi)$, and if $x \in U \cap V$ and $F \rightsquigarrow x$, then also $x \in U$ and $x \in V$ and we conclude $U, V \in F$, whence $U \cap V \in F$ since $F$ is an ultrafilter, so that $U \cap V$ satisfies the condition of belonging to $\tau(\xi)$. Given a collection of elements $U_i \in \tau(\xi)$, if $x \in \cup_i U_i$ and $F \rightsquigarrow x$, then $x \in U_i$ for some $i$ and we conclude $U_i \in F$, whence $\cup_i U_i \in F$ since $F$ is upward closed. Therefore $\cup_i U_i$ satisfies the condition of belonging to $\tau(\xi)$ (vacuously so if the collection is empty). \end{proof} \begin{prop} \label{galois}\hypertarget{galois}{} There is a Galois connection between notions of convergence on $S$ and subsets of $P(S)$, according to the bi-implication \begin{displaymath} \mathcal{C} \subseteq \tau(\xi) \; \Leftrightarrow \; \xi \subseteq conv(\mathcal{C}). \end{displaymath} \end{prop} \begin{proof} To establish the bi-implication, it suffices to observe that both containments $\mathcal{C} \subseteq \tau(\xi)$ and $\xi \subseteq conv(\mathcal{C})$ are equivalent to the condition \begin{displaymath} \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). \end{displaymath} \end{proof} \begin{prop} \label{fix}\hypertarget{fix}{} If $\mathcal{O}$ is a topology on $S$, then $\mathcal{O} = \tau(conv(\mathcal{O}))$ (i.e., topologies are fixed points of the [[closure operator]] $\tau \circ \conv$). \end{prop} \begin{proof} We already have $\mathcal{O} \subseteq \tau(conv(\mathcal{O}))$ from Proposition \ref{galois}. For the other direction, we must show that any $V$ belonging to $\tau(conv(\mathcal{O}))$ is an $\mathcal{O}$-neighborhood of each of its points. Suppose the contrary: that $x \in V$ but $V$ is not an $\mathcal{O}$-neighborhood of $x$. Holding $V$ fixed, for every $\mathcal{O}$-neighborhood $U \in N_x$ we have $U \cap \neg V \neq \emptyset$, so that sets of the form $U \cap \neg V$ with $U$ ranging over $N_x$ generate a filter. By the ultrafilter principle, we may extend this filter to an ultrafilter $F$; clearly we have $F \rightsquigarrow x$ and $\neg V \in F$, but since $F \rightsquigarrow x$ and $V \in \tau(conv(\mathcal{O}))$ and $x \in V$, we also have $V \in F$, which is inconsistent with $\neg V \in F$. \end{proof} Propositions \ref{topology}, \ref{galois}, and \ref{fix} more or less show that a topological space $(S, \mathcal{O})$ is a particular type of [[pseudotopological space]]: \begin{defn} \label{}\hypertarget{}{} A pseudotopological space is set $S$ equipped with a relation $\xi: \beta S \to S$ such that $1_S \leq \xi \circ u_S$. \end{defn} All that remains is to check is: \begin{prop} \label{pseudo}\hypertarget{pseudo}{} The lax unit condition $1_S \leq \xi \circ u_S$ holds if $\xi = conv(\mathcal{O})$, for a topology $\mathcal{O}$. \end{prop} The unit $u_S: S \to \beta S$ may also be denoted $prin_S$, as it takes an element $x \in S$ to the principal ultrafilter \begin{displaymath} prin_S(x) = \{U \subseteq S: x \in U\} \end{displaymath} and now the unit condition says $prin_S(x) \rightsquigarrow_\xi x$ for all $x$. For $\xi = conv(\mathcal{O})$, this says $N_x \subseteq prin_S(x)$, or that $x \in V$ for all neighborhoods $V \in N_x$, which is a tautology. One of our goals is to prove the following theorem: \begin{theorem} \label{conc}\hypertarget{conc}{} \textbf{(Main Theorem)} An arrow $\xi: \beta(S) \to S$ in $Rel$ is of the form $conv(\tau(\xi))$ if and only if the following inequalities are satisfied: \begin{displaymath} 1_S \leq \xi \circ prin_S, \qquad \xi \circ \beta(\xi) \leq \xi \circ m_S \end{displaymath} where $m_S: \beta \beta(S) \to \beta(S)$ is the multiplication on the ultrafilter monad. \end{theorem} \hypertarget{functor}{}\subsection*{{Ultrafilter monad on $Rel$}}\label{functor} 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 $\beta$-module, leading up to the context of generalized multicategories. \hypertarget{extending_the_ultrafilter_functor_to_}{}\subsubsection*{{Extending the ultrafilter functor to $Rel$}}\label{extending_the_ultrafilter_functor_to_} First we examine more closely the extension of the ultrafilter functor $\beta: Set \to Set$ to $Rel$, showing in particular that the extension is a strict functor. First we slightly rephrase our earlier definition: \begin{defn} \label{}\hypertarget{}{} For a relation $r: X \to Y$ between sets, given by a subobject $R \hookrightarrow X \times Y$ in $Set$ with projections $f: R \to X$ and $g: R \to Y$, define $\bar{\beta}(r)$ to be the composite \begin{displaymath} \beta(X) \stackrel{\beta(f)^{o}}{\to} \beta(R) \stackrel{\beta(g)}{\to} \beta(Y) \end{displaymath} in the bicategory of relations. \end{defn} Any span of functions $(h: S \to X, k: S \to Y)$ that represents $r$ (in the sense that $r = k h^{o}$ in the bicategory of relations) would serve in place of $(f, g)$, since for any such span there is an epi $s: S \to R$ with $h = f s$, $k = g s$, whence $\beta(s)$ is epi (because the epi $s$ splits in $Set$) and we have \begin{displaymath} \itexarray{ \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}. } \end{displaymath} In particular, $\bar{\beta}$ is well-defined. Since $\bar{\beta}$ extends $\beta: Set \to Set$, there is no harm in writing $\beta(r)$ in place of $\bar{\beta}(r)$. If $r \leq r': X \to Y$, then $\beta(r) \leq \beta(r')$ (as can be seen from the calculation displayed above, but replacing the epi $s$ by a general map $t$, and the first equation by an inequality $\geq$). \begin{remark} \label{oplax}\hypertarget{oplax}{} The same recipe works to extend any functor $T: Set \to Set$ to $Rel$, and the extension $\bar{T}: Rel \to Rel$ is always an \emph{op-lax} functor in the sense that \begin{displaymath} \bar{T}(r s) \leq \bar{T}(r) \bar{T}(s) \end{displaymath} as is easily seen by contemplating a pullback diagram (where $r = g f^{o}$ and $s = k h^{o}$): \begin{displaymath} \itexarray{ & & & & Q & & & & \\ & & & \mathllap{p} \swarrow & & \searrow \mathrlap{q} & & & \\ & & R & & & & S & & \\ & \mathllap{f} \swarrow & & \searrow \mathrlap{g} & & \mathllap{h} \swarrow & & \searrow \mathrlap{k} & \\ X & & & & Y & & & & Z } \end{displaymath} whereupon one calculates \begin{displaymath} \itexarray{ \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) } \end{displaymath} where the inequality comes from $T(q) T(p)^{o} \leq T(h)^{o} T(g)$, which is equivalent to $T(h) T(q) \leq T(g) T(p)$ (where even equality holds). This calculation shows that $\bar{T}$ is an actual (not just an op-lax) functor on $Rel$ iff $T$ satisfies the \textbf{Beck-Chevalley} condition: if $(p, q)$ is a pullback of $(g, h)$, then \begin{displaymath} T(q) T(p)^{o} = T(h)^{o} T(g). \end{displaymath} This in turn amounts to $T$ preserving weak pullbacks. (It actually says $T$ takes pullbacks to weak pullbacks, but this implies $T$ takes weak pullbacks to weak pullbacks because any endofunctor $T$ on $Set$ preserves epis, using the axiom of choice.) \end{remark} \begin{prop} \label{BC}\hypertarget{BC}{} The functor $\beta: Set \to Set$ satisfies the Beck-Chevalley condition (and therefore the extension $\beta: Rel \to Rel$ is a strict functor). \end{prop} \begin{proof} Referring to the pullback diagram in Remark \ref{oplax}, let $Q = R \times_Y S$ be the pullback. We must show that the canonical map \begin{displaymath} \beta(R \times_Y S) \to \beta(R) \times_{\beta(Y)} \beta(S) \end{displaymath} is epic. Viewing this as a continuous map between compact Hausdorff spaces (see \href{http://ncatlab.org/nlab/show/compactum#ultrafilters_form_a_compactum_23}{this section} of the article on [[compacta]]), it suffices to show that the canonical map \begin{displaymath} R \times_Y S \to \beta(R) \times_{\beta(Y)} \beta(S) \end{displaymath} has a dense image. Let $(G, H) \in \beta(R) \times_{\beta(Y)} \beta(S)$, so that $\beta(g)(G) = \beta(h)(H)$ are the same ultrafilter $J \in \beta(Y)$. Let $\hat{A}$ and $\hat{B}$ be \href{http://ncatlab.org/nlab/show/compactum#the_space_of_ultrafilters_19}{basic open neighborhoods} of $G$ and $H$ in $\beta(R)$ and $\beta(S)$ respectively; we must show that there is $(r, s) \in R \times_Y S$ such that \begin{displaymath} (prin(r), prin(s)) \in \hat{A} \times \hat{B} \end{displaymath} or in other words such that $r \in A$ and $s \in B$. We have $g^{-1}(g(A)) \in G$ since $A \in G$ and $A \subseteq g^{-1}(g(A))$, so that $g(A)$ belongs to \begin{displaymath} J = \beta(g)(G) \coloneqq \{C \subseteq Y: g^{-1}(C) \in G\} \end{displaymath} and similarly $h(B) \in J$. It follows that $g(A) \cap h(B) \in J$ so that $g(A) \cap h(B) \neq \emptyset$. Any element $y \in g(A) \cap h(B)$ can be written as $y = g(r)$ and $y = h(s)$ for some $r \in A$ and $s \in B$, and this completes the proof. \end{proof} \hypertarget{ultrafilter_monad_on_the_equipment_}{}\subsubsection*{{Ultrafilter monad on the equipment $\mathbf{Rel}$}}\label{ultrafilter_monad_on_the_equipment_} As mentioned in an \href{/nlab/show/relational+beta-module#abstract}{earlier section}, the natural transformations $u = prin: 1_{Set} \to \beta$, $m: \beta\beta \to \beta$ \textbf{do not} extend to (strict) natural transformations on the locally posetal bicategory $Rel$, but only to transformations that are op-lax in the sense of inequalities \begin{displaymath} prin_Y \circ r \leq \beta(r) \circ prin_X, \qquad m_Y \circ \beta\beta(r) \leq \beta(r) \circ m_X \end{displaymath} for every relation $r: X \to Y$. These are equivalent to inequalities \begin{displaymath} r \leq prin_Y^o \circ \beta(r) \circ prin_X, \qquad \beta\beta(r) \leq m_Y^o \circ \beta(r) \circ m_X \end{displaymath} and they may be deduced simply by staring at naturality diagrams in $Set$, in which we represent or [[bicategory of relations|tabulate]] $r$ by $\pi_2 \circ \pi_1^o$: \begin{displaymath} \itexarray{ & & 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) } \end{displaymath} \textbf{To get an actual monad}, it is more satisfactory in this context to consider not the bicategory $Rel$, but rather the [[equipment]] or [[framed bicategory]] $\mathbf{Rel}$. That is, there is a 2-category $Equip$ 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 $\bar{\beta}: \mathbf{Rel} \to \mathbf{Rel}$. In more detail: the 0-cells of $\mathbf{Rel}$ are sets, and the horizontal arrows are relations between sets. Vertical arrows are functions between sets, and a 2-cell of shape \begin{displaymath} \itexarray{ A & \stackrel{r}{\to} & B \\ \mathllap{f} \downarrow & \Downarrow & \downarrow \mathrlap{g} \\ C & \stackrel{s}{\to} & D; } \end{displaymath} is an inequality $g \circ r \leq s \circ f$. We straightforwardly get a [[double category]] $\mathbf{Rel}$, and the ultrafilter functor on $Set$ extends to a functor $\bar{\beta}: \mathbf{Rel} \to \mathbf{Rel}$ 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 $F, G: \mathbf{B} \to \mathbf{C}$ between equipments. A \emph{transformation} $\eta: F \to G$ assigns to each 0-cell $b$ of $\mathbf{B}$ a \emph{vertical} arrow $\eta b: F b \to G b$, and to each horizontal arrow $r: b \to b'$ a 2-cell $\eta r$ of the form \begin{displaymath} \itexarray{ 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'; } \end{displaymath} suitably compatible with the double category structures. We thus find that the op-lax structures of the transformations $prin: 1 \to \bar{\beta}$, $m: \bar{\beta} \bar{\beta} \to \bar{\beta}$ on $Rel$ qua \emph{bicategory} are exactly what we need to produce honest transformations $u: 1 \to \bar{\beta}$, $m: \bar{\beta} \bar{\beta} \to \bar{\beta}$ on $\mathbf{Rel}$ qua \emph{equipment}, and the result is an ultrafunctor monad on the equipment $\mathbf{Rel}$. Given a monad $T$ on an equipment $\mathbf{B}$, one may proceed to construct a \emph{horizontal Kleisli equipment} $HKl(\mathbf{B}, T)$ with the same 0-cells and vertical arrows as $\mathbf{B}$, but whose horizontal arrows are of the form $r: b \to T b'$. A 2-cell in $HKl(\mathbf{B}, T)$ (with vertical source $f$ and vertical target $g$) is a 2-cell in $\mathbf{B}$ of the form \begin{displaymath} \itexarray{ b & \stackrel{r}{\to} & T b' \\ \mathllap{f} \downarrow & \Downarrow \mathrlap{\alpha} & \downarrow \mathrlap{T g} \\ c & \stackrel{s}{\to} & T c'; } \end{displaymath} 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 $m_b: T T b \to T b$, 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 \emph{monoid} consists of a horizontal arrow $\xi: b \to b$ together with unit and multiplication 2-cells \begin{displaymath} \itexarray{ 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 } \end{displaymath} satisfying evident identities. A \emph{monoid homomorphism} from $(b, \xi)$ to $(c, \theta)$ consists of a vertical arrow and 2-cell $(f, \phi)$ of the form \begin{displaymath} \itexarray{ b & \stackrel{\xi}{\to} & b \\ \mathllap{f} \downarrow & \Downarrow \mathrlap{\phi} & \downarrow \mathrlap{f} \\ c & \underset{\theta}{\to} & c } \end{displaymath} that is suitably compatible with the unit and multiplication cells. The following notion gives an \emph{interim} notion of generalized multicategory that applies in particular to relational $\beta$-modules. \begin{defn} \label{}\hypertarget{}{} Given a monad $T$ on an equipment $\mathbf{B}$, a $T$-\emph{monoid} is a monoid in the horizontal Kleisli equipment $HKl(\mathbf{B}, T)$. A \emph{map} of $T$-monoids is a homomorphism between monoids in $HKl(\mathbf{B}, T)$. \end{defn} \begin{prop} \label{}\hypertarget{}{} For the ultrafilter monad $\beta$ on the equipment $\mathbf{Rel}$, a structure of $\beta$-monoid is equivalent to a structure of relational $\beta$-module, and a homomorphism of $\beta$-monoids is the same as a lax map of relational $\beta$-modules in the bicategory $Rel$. \end{prop} \begin{proof} This is really just a matter of unwinding definitions. The data of a $\beta$-monoid in the equipment $\mathbf{Rel}$ amounts to a set $X$ together with a horizontal arrow in the Kleisli construction, that is to say a relation $c: X \to \beta X$ (opposite to our conventional direction, i.e., $c = \xi^o$). The unit and multiplication cells for $c$ are inequalities $1_X \leq c$ and $c \circ_{Kl} c \leq c$ (the vertical source and target being identity maps), where the identity $1_X$ in the Kleisli construction uses the unit for $\beta$ and the Kleisli composition uses the multiplication. Back in the bicategory $Rel$ these translate to relational inequalities \begin{displaymath} prin_X \leq c, \qquad m_X \circ \beta c \circ c \leq c \end{displaymath} or, with $c = \xi^o$, \begin{displaymath} prin_X \leq \xi^o, \qquad m_X \circ (\beta (\xi))^o \circ \xi^o \leq \xi^o. \end{displaymath} These boil down to relational inequalities \begin{displaymath} 1_X \leq prin_X^o \circ \xi^o, \qquad (\beta (\xi))^o \circ \xi^o \leq m_X^o \circ \xi^o \end{displaymath} or to \begin{displaymath} 1_X \leq \xi \circ prin_X, \qquad \xi \circ \beta (\xi) \leq \xi \circ m_X, \end{displaymath} as in the axioms on relational beta-modules. Similarly, a $\beta$-monoid homomorphism $(X, c) \to (Y, d)$ is a vertical arrow $f: X \to Y$ in $HKl(\mathbf{Rel}, \beta)$ together with a suitable 2-cell, which after some unraveling comes down to a relational inequality \begin{displaymath} \beta (f) \circ c \leq d \circ f \end{displaymath} or to an inequality $\beta (f) \circ \xi_X^o \leq \xi_Y^o \circ f$, which may be further massaged into the form $\xi_X^o \circ f^o \leq (\beta (f))^o \circ \xi_Y^o$, or simply to \begin{displaymath} f \circ \xi_X \leq \xi_Y \circ \beta (f) \end{displaymath} as advertised in the notion of lax morphism of relational $\beta$-modules (cf. theorem \ref{contmap} below). \end{proof} \begin{remark} \label{}\hypertarget{}{} In some sense, relational $\beta$-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 $T$ acting on \emph{virtual} equipments and study \emph{normalized} $T$-monoids. Explaining all this requires a lengthy build-up. Even in the relatively restricted packet of unifications that come under the rubric of $(T, V)$-algebras, as studied by \hyperlink{CHT2}{Clementino, Hofmann, Tholen}, \hyperlink{Seal}{Seal} and others as a way of bringing [[topological space|topological spaces]], [[uniform space|uniform spaces]], [[metric space|metric spaces]], [[approach space|approach spaces]], [[Moore closure|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 $V$-matrices) can be somewhat elaborate, and mildly daunting. The example of the ultrafilter monad acting on $Rel$ 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. \end{remark} \hypertarget{proof_of_main_theorem}{}\subsection*{{Proof of Main Theorem}}\label{proof_of_main_theorem} We now return to the task of proving theorem \ref{conc}. \begin{remark} \label{open}\hypertarget{open}{} For a topological space $(S, \mathcal{O})$ and a point $x \in S$, let $\xi = conv(\mathcal{O})$, and let $\mathcal{O}_x$ be the collection of \emph{open} neighborhoods of $x$. Then $F \rightsquigarrow_\xi x$, i.e., $N_x \subseteq F$, is equivalent to $\mathcal{O}_x \subseteq F$. This is because $N_x$ is the filter generated by $\mathcal{O}_x$. \end{remark} \begin{remark} \label{}\hypertarget{}{} The following conditions are equivalent: \begin{itemize}% \item $\xi = conv(\mathcal{C})$ for some $\mathcal{C} \subseteq P(S)$; \item $\xi = conv(\mathcal{O})$ for some topology $\mathcal{O} \subseteq P(S)$; \item $\xi = conv(\tau(\xi))$. \end{itemize} Indeed, $conv \circ \tau \circ conv = conv$ by general properties of Galois connections. Applying $conv \circ \tau$ to both sides of the first equation, we have \begin{displaymath} conv(\tau(\xi)) = (conv \circ \tau \circ conv)(\mathcal{C}) = conv(\mathcal{C}) = \xi \end{displaymath} so the first equation implies the third. Which in turn implies the second, since we know by proposition \ref{topology} that collections $\mathcal{O}$ of the form $\tau(\xi)$ are topologies. The second equation trivially implies the first. \end{remark} We now break up our Main Theorem \ref{conc} into the following two theorems. \begin{theorem} \label{}\hypertarget{}{} If $\xi = conv(\mathcal{O})$ for a topology $\mathcal{O}$, then the two inequalities of \eqref{rel1} are satisfied. \end{theorem} \begin{proof} The first inequality (lax unit condition) was already verified in proposition \ref{pseudo}. For the second (lax associativity), let us represent the relation $\xi$ by a span $\beta S \stackrel{\pi_1}{\leftarrow} R \stackrel{\pi_2}{\to} S$, so that $\beta(\xi) = \beta(\pi_2) \beta(\pi_1)^o$. The lax associativity condition becomes \begin{displaymath} \pi_2 \pi_1^o \beta(\pi_2) \beta(\pi_1)^o \leq \pi_2 \pi_1^o m_S \end{displaymath} which (using $\beta(\pi_1) \dashv \beta(\pi_1)^o$) is equivalent to \begin{equation} \pi_2 \pi_1^o \beta(\pi_2) \leq \pi_2 \pi_1^o m_S \beta(\pi_1) \label{assoc}\end{equation} or in other words that for all $\mathcal{G}: \beta(R)$, $x: S$ \begin{equation} \beta(\pi_2)(\mathcal{G}) \rightsquigarrow_\xi x \;\; \vdash \;\; m_S \beta(\pi_1)(\mathcal{G}) \rightsquigarrow_\xi x. \label{conv1}\end{equation} Here $\beta(\pi_2)(\mathcal{G})$ is, by definition, \begin{displaymath} \{U \subseteq S: \pi_2^{-1}(U) \in \mathcal{G}\}, \end{displaymath} with $\beta(\pi_1)(\mathcal{G})$ defined similarly. The monad multiplication $m_S: \beta \beta S \to \beta S$ is by definition \begin{displaymath} (\mathcal{U}: \beta\beta S) \;\; m_S(\mathcal{U}) \coloneqq \{A \subseteq S: \hat{A} \in \mathcal{U}\} \end{displaymath} where $\hat{A} = \{F \in \beta S: A \in F\}$ (see also the \href{/nlab/show/relational+beta-module#functor}{previous section}). Thus, \eqref{conv1} translates into the following entailment (using remark \ref{open}): \begin{displaymath} \itexarray{ & & \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}. } \end{displaymath} This would naturally follow if \begin{displaymath} \forall_{U \in \mathcal{O}_x} \pi_2^{-1}(U) \subseteq \pi_1^{-1}(\hat{U}). \end{displaymath} But a pair $(F, y)$ belongs to $\pi_2^{-1}(U)$ if $F \rightsquigarrow_\xi y$ and $y \in U$; we want to show this implies $F = \pi_1(F, y)$ belongs to $\hat{U}$, or in other words that $U \in F$. But this is tautological, given how $conv(\mathcal{O})$ is defined in terms of a topology $\mathcal{O}$. \end{proof} The next theorem establishes the converse of the preceding theorem; the two theorems together establish the Main Theorem. First we need a remark and a lemma. \begin{remark} \label{closed0}\hypertarget{closed0}{} Given any relation $\xi: Rel(\beta S, S)$ and $A \subseteq S$, then $A$ is closed wrt the topology $\tau(\xi)$ if and only if for all $x \in S$, \begin{displaymath} (\exists_{F: \beta S} A \in F \; \wedge \; F \rightsquigarrow_\xi x) \implies x \in A \end{displaymath} This follows by inverting the definition of the open sets in $\tau(\xi)$. \end{remark} \begin{lemma} \label{closed}\hypertarget{closed}{} If a relation $\xi: Rel(\beta S, S)$ satisfies the inequalities of \eqref{rel1} and $x: S$, $A \subseteq S$, we have that $x$ belongs to the closure $\bar{A}$ wrt the topology $\tau(\xi)$ if and only if $\exists_{F: \beta S} A \in F \; \wedge \; F \rightsquigarrow_\xi x$. \end{lemma} \begin{proof} For any $A \subseteq S$, denote $A^+ = \{x \mid \exists_{F: \beta S} A \in F \; \wedge \; F \rightsquigarrow_\xi x\}$; we want to show that $A^+ = \bar{A}$. It suffices to show that $A \subseteq A^+ \subseteq \bar{A}$ and that $A^+$ is closed. That $A^+ \subseteq \bar{A}$ is clear from the characterization of closed sets given in Remark \ref{closed0}; $A^+$ is in some sense the ``one-step closure'' of $A$. That $A \subseteq A^+$ follows from the lax unit condition for relational $\beta$-modules: if $x \in A$, then $prin(x) \rightsquigarrow_\xi x$ and $A \in prin(x)$. It's clear from the characterization of closed sets given in Remark \ref{closed0} that a set $A$ is closed if and only if $A = A^+$. We will establish that $A^+$ is closed by showing that $(A^+)^+ = A^+$. By the previous paragraph we know that $A^+ \subseteq (A^+)^+$ so we just need the reverse containment. So suppose that $x \in (A^+)^+$, and pick an ultrafilter $F \in \beta S$ with $A^+ \in F$ and $F \rightsquigarrow_\xi x$. In order to show that $x \in A^+$, we need to produce an ultrafilter $F' \in \beta S$ with $A \in F'$ such that $F' \rightsquigarrow_\xi x$. We will do this by applying the lax associativity condition, using an appropriate ultrafilter $\mathcal{G} \in \beta R$. In fact, we claim that any $\mathcal{G}$ extending the following \href{http://ncatlab.org/nlab/show/filter#filterbases}{filterbase} on $R$: \begin{displaymath} \mathcal{G}_0 = \{\pi_1^{-1}(\hat{A}) \cap \pi_2^{-1}(U) \mid U \in F\} \end{displaymath} will fit the bill. First let us verify that such an ultrafilter $\mathcal{G}$ exists. By the ultrafilter principle, it suffices to verify that $\mathcal{G}_0$ generates a proper filter. It's clear that $\mathcal{G}_0$ is closed under finite intersection. So the filter it generates is proper iff $\mathcal{G}_0$ is proper, i.e. doesn't contain the empty set. Now, a typical element of $\mathcal{G}_0$ is of the form $\pi_1^{-1}(\hat{A}) \cap \pi_2^{-1}(U) = \{(G \in \beta S, \; y \in U) \mid A \in G, G \rightsquigarrow_\xi y\}$ for some $U \in F$. Since $U \in F$ and $A^+ \in F$, we can pick $y \in U \cap A^+$. Since $y \in A^+$, there is $G \in \beta S$ with $A \in G$ such that $G \rightsquigarrow_\xi y$ as desired. So we can pick a $\mathcal{G} \in \beta R$ extending $\mathcal{G}_0$. We want to establish that \begin{enumerate}% \item $\beta \pi_2(\mathcal{G}) \rightsquigarrow_\xi x$ \item $A \in F':= m_S(\beta \pi_1(\mathcal{G}))$ \end{enumerate} This will complete the proof: From (1) it follows that $F' \rightsquigarrow_\xi x$ by lax associativity. Then (2), along with the fact that $F' \rightsquigarrow_\xi x$, implies that $x \in A^+$, as desired. To show (1) we show that $\beta \pi_2(\mathcal{G}) = F$; this suffices since $F \rightsquigarrow_\xi x$ by hypothesis. Since both sides of the equations are ultrafilters, It further suffices to show that $F \subseteq \beta \pi_2(\mathcal{G})$. So suppose that $U \in F$. We want to show that $U \in \beta \pi_2(\mathcal{G})$ i.e. that $\pi_2^{-1}(U) \in \mathcal{G}$. This is true because $\pi_1^{-1}(\hat{A}) \cap \pi_2^{-1}(U) \in \mathcal{G}$ and the fact that $\mathcal{G}$ is upward closed. For (2) we want to show that $A \in m_S(\beta \pi_1(\mathcal{G}))$ i.e. that $\hat{A} \in \beta \pi_1(\mathcal{G})$, i.e. that $\pi_1^{-1}(\hat{A}) \in \mathcal{G}$. This is true because $\pi_1^{-1}(\hat{A}) \cap \pi_2^{-1}(U) \in \mathcal{G}$ and the fact that $\mathcal{G}$ is upward closed. \end{proof} \begin{theorem} \label{}\hypertarget{}{} If $\xi: \beta S \to S$ in $Rel$ satisfies the inequalities of \eqref{rel1}, then $\xi = conv(\tau(\xi))$. \end{theorem} \begin{proof} We have $\xi \leq conv(\tau(\xi))$ from the Galois connection (proposition \ref{galois}), so we just need to prove $conv(\tau(\xi)) \leq \xi$, or that $F \rightsquigarrow_{conv(\tau(\xi))} x$ (henceforth abbreviated as $F \rightsquigarrow_{\tau(\xi)} x$) implies $F \rightsquigarrow_\xi x$ under the conditions \eqref{rel1}. If $F \rightsquigarrow_{\tau(\xi)} x$, then every neighborhood $V$ of $x$ belongs to $F$, so that for every $U \in F$, every neighborhood $V$ of $x$ intersects $U$ in a nonempty set. But this just means $x \in \bar{U}$ for every $U \in F$, or in other words (using lemma \ref{closed}) that \begin{displaymath} U \in F \; \; \vdash \; \; \exists_{G: \beta S} U \in G \; \wedge \; G \rightsquigarrow_\xi x. \end{displaymath} Representing the relation $\xi$ as usual by a subset $\langle \pi_1, \pi_2 \rangle: R \hookrightarrow \beta S \times S$, another way of expressing the existential formula on the right of this entailment is: \begin{displaymath} \exists_{(G, x): \beta S \times S} \; (G, x) \in R \; \wedge \; G \in \hat{U} \end{displaymath} or \begin{displaymath} \exists_{\gamma: R} \pi_1(\gamma) \in \hat{U} \wedge \pi_2(\gamma) = x \end{displaymath} or even just \begin{equation} \pi_1^{-1}(\hat{U}) \wedge \pi_2^{-1}(\{x\}) \neq \emptyset \label{ult}\end{equation} as subsets of $R$, as $U$ ranges over all elements of $F$. We therefore have that subsets of the form \eqref{ult} generate a proper filter of $R$. By the ultrafilter principle, we may extend this filter to an ultrafilter $\mathcal{G} \in \beta R$. By construction, we have \begin{displaymath} 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}\} \end{displaymath} but in fact these inclusions are equalities since the left sides and right sides are ultrafilters. Put differently, we have established \begin{displaymath} F = (m_S \circ \beta(\pi_1))(\mathcal{G}), \qquad prin_S(x) = \beta(\pi_2)(\mathcal{G}). \end{displaymath} Notice that the lax unit condition of \eqref{rel1} implies that $prin_S(x) = \beta(\pi_2)(\mathcal{G}) \rightsquigarrow_\xi x$, or that $(\mathcal{G}, x)$ belongs to the relation $\xi \circ \beta(\pi_2)$. Recall also that the lax associativity condition is equivalent to \eqref{assoc}, which says \begin{displaymath} \xi \circ \beta(\pi_2) \leq \xi \circ m_S \circ \beta(\pi_1); \end{displaymath} in other words $(\mathcal{G}, x)$ belongs to $\xi \circ m_S \circ \beta(\pi_1)$, i.e., $F = (m_S \circ \beta(\pi_1))(\mathcal{G}) \rightsquigarrow_\xi x$, as was to be shown. \end{proof} This completes the proof of the Main Theorem (theorem \ref{conc}). \hypertarget{continuous_maps}{}\subsection*{{Continuous maps}}\label{continuous_maps} \begin{theorem} \label{contmap}\hypertarget{contmap}{} A function between two topological spaces $f: X \to Y$ is continuous if and only if $f \circ \xi \leq \theta \circ \beta(f)$ for their respective topological notions of convergence $\xi, \theta$. \end{theorem} \begin{proof} Suppose first that $f$ is continuous, and that $(F, y) \in \beta(X) \times Y$ belongs to $f \circ \xi$, i.e., there is $x$ such that $F \rightsquigarrow x$ and $f(x) = y$. We want to show $\beta(f)(F) \rightsquigarrow y = f(x)$, or that any open set $V$ containing $f(x)$ belongs to $\beta(f)(F)$. The latter means $f^{-1}(V) \in F$, which is true since $f^{-1}(V)$ is an open set containing $x$ and $F \rightsquigarrow x$. Now suppose $f \circ \xi \leq \theta \circ \beta(f)$. To show $f$ is continuous, it suffices to show that \begin{displaymath} f(\bar{A}) \subseteq \widebar{f(A)} \end{displaymath} for any $A \subseteq X$ (easy exercise). For $x \in \bar{A}$, lemma \ref{closed} shows there is $F: \beta(X)$ with $A \in F$ and $F \rightsquigarrow x$. Under the supposition we have $\beta(f)(F) \rightsquigarrow f(x)$, and we also have $f(A) \in \beta(f)(F)$, because $A \subseteq f^{-1}(f(A))$ and $F$ is upward closed and $A \in F$ implies $f^{-1}(f(A)) \in F$. Then again by lemma \ref{closed}, $f(A) \in \beta(f)(F)$ and $\beta(f)(F) \rightsquigarrow f(x)$ implies $f(x) \in \widebar{f(A)}$, as desired. \end{proof} \begin{cor} \label{equivalence}\hypertarget{equivalence}{} \textbf{(Barr)} The category of topological spaces is equivalent (even isomorphic to) the category of lax $\beta$-modules and lax morphisms between them. \end{cor} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} As above, a [[subset]] $A$ of $S$ is \textbf{[[open subset|open]]} if $A \in \mathcal{U}$ whenever $\mathcal{U} \to x \in A$. On the other hand, by lemma \ref{closed}, $A$ is \textbf{[[closed subset|closed]]} if $x \in A$ whenever $A \in \mathcal{U} \to x$. A relational $\beta$-module is \textbf{[[compact space|compact]]} if every ultrafilter converges to at least one point. It is \textbf{[[Hausdorff space|Hausdorff]]} if every ultrafilter converges to at most one point. Thus, a \textbf{[[compactum]]} is (assuming $UF$) 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 $Set$ rather than in $Rel$. Full proofs may be found at [[compactum]]; see also [[ultrafilter monad]]. A continuous map $f$ from $(X, \xi: \beta X \to X)$ to $(Y, \theta: \beta Y \to Y)$ is [[proper map|proper]] if the square \begin{displaymath} \itexarray{ \beta X & \stackrel{\xi}{\to} & X \\ \mathllap{\beta (f)} \downarrow & & \downarrow \mathrlap{f} \\ \beta Y & \underset{\theta}{\to} & Y } \end{displaymath} commutes (strictly) in $Rel$, and $f$ is [[open map|open]] if the square \begin{displaymath} \itexarray{ \beta X & \stackrel{\xi}{\to} & X \\ \mathllap{\beta (f)^o} \uparrow & & \uparrow \mathrlap{f^o} \\ \beta Y & \underset{\theta}{\to} & Y } \end{displaymath} commutes in $Rel$. From this point of view, a space $X$ is Hausdorff if the diagonal map $\delta: X \to X \times X$ is open, and compact if $\epsilon: X \to 1$ is proper (and these facts remain true even for pseudotopological spaces). See \hyperlink{CHJ}{Clementino, Hofmann, and Janelidze}, \emph{infra} corollary 2.5. The following \emph{ultrafilter interpolation} result is due to \hyperlink{Pis}{Pisani}: \begin{utheorem} A topological space $(X, \xi)$ is exponentiable if, whenever $m_X(\mathcal{U}) \rightsquigarrow_\xi x$ for $\mathcal{U} \in \beta\beta X$ and $x \in X$, there exists $F \in \beta X$ with $\mathcal{U} \rightsquigarrow_{\beta(\xi)} F$ and $F \rightsquigarrow_\xi x$. \end{utheorem} For an convergence-approach extension of this result to exponentiable \emph{maps} in $Top$, see \hyperlink{CHT}{Clementino, Hofmann, and Tholen}. \begin{defn} \label{}\hypertarget{}{} A continuous map $f: X \to Y$ is a \emph{discrete fibration} if, whenever $G \rightsquigarrow y$ in $Y$ and $f(x) = y$, there exists a unique $F \in \beta(X)$ such that $\beta(f)(F) = G$ and $F \rightsquigarrow x$ in $X$. \end{defn} \begin{prop} \label{}\hypertarget{}{} A continuous map $f: X \to Y$ is \'e{}tale (a local homeomorphism) if $f$ and $\delta_f: X \to X \times_Y X$ are both discrete fibrations. \end{prop} For more on this, see \hyperlink{CHJ}{Clementino, Hofmann, and Janelidze}. \hypertarget{relation_to_nonstandard_analysis}{}\subsection*{{Relation to nonstandard analysis}}\label{relation_to_nonstandard_analysis} In [[nonstandard analysis]] (which implicitly relies throughout on $UF$), 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 \textbf{[[standard part]]} of $u$ or that $u$ belongs to the \textbf{[[halo]]} (or \emph{monad}, but \emph{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 \emph{the} standard part of $u$ only for Hausdorff spaces.) So ultrafilters behave very much like hyperpoints. This is not to say that ultrafilters \emph{are} (or even can be) hyperpoints, as they don't obey the [[transfer principle]]. Nevertheless, one does use ultrafilters to construct the [[model]]s 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. \hypertarget{relation_to_other_topological_concepts}{}\subsection*{{Relation to other topological concepts}}\label{relation_to_other_topological_concepts} If $\beta$ is treated as a monad on $Set$ instead of on $Rel$, then its algebras are the [[compacta]] (the compact Hausdorff spaces), again assuming $UP$; 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 space|uniform structures]], [[prometric spaces]], and [[approach space|approach structures]], see \hyperlink{CHT2}{Clementino, Hofmann, and Tholen}. \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item Michael Barr, \emph{Relational algebras}, Springer Lecture Notes in Math. 137 (1970), 39-55. \end{itemize} \begin{itemize}% \item Maria Manuel Clementino, Dirk Hofmann, and George Janelidze, \emph{On exponentiability of \'e{}tale algebraic homomorphisms}, Preprint 11-35, University of Coimbra. (\href{http://www.mat.uc.pt/preprints/ps/p1135.pdf}{pdf}) \end{itemize} \begin{itemize}% \item Maria Manuel Clementino, Dirk Hofmann, and Walter Tholen, \emph{The convergence approach to exponentiable maps}, Portugaliae Mathematica (Nova Series), Vol. 60 Issue 2 (2003), 139-160. (\href{https://eudml.org/doc/51942}{web}) (\href{https://cmuc.mat.uc.pt/rdonweb/publications/publication22.pdf}{pdf}) \end{itemize} \begin{itemize}% \item Maria Manuel Clementino, Dirk Hofmann, and Walter Tholen, \emph{One Setting for All: Metric, Topology, Uniformity, Approach Structure.} (\href{http://www.math.yorku.ca/~tholen/ProMat_V_.pdf}{pdf}) \end{itemize} \begin{itemize}% \item Gavin J. Seal, \emph{Canonical and op-canonical lax algebras}, Theory and Applications of Categories, 14 (2005), 221--243. (\href{http://www.tac.mta.ca/tac/volumes/14/10/14-10abs.html}{web}) \end{itemize} \begin{itemize}% \item Claudio Pisani, Convergence in exponentiable spaces, Theory Appl. Categories 5 (1999), 148-162. (\href{http://www.tac.mta.ca/tac/volumes/1999/n6/5-06abs.html}{web}) \end{itemize} \begin{itemize}% \item [[Geoff Cruttwell]] and [[Mike Shulman]], \emph{A unified framework for generalized multicategories}, \href{http://arxiv.org/abs/0907.2460}{arxiv/0907.2460} \end{itemize} [[!redirects relational beta-module]] [[!redirects relational beta-modules]] [[!redirects relational ∞-module]] [[!redirects relational ∞-modules]] \end{document}