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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*{prometric space} \hypertarget{prometric_spaces}{}\section*{{Prometric spaces}}\label{prometric_spaces} \noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak \noindent\hyperlink{elementary}{Elementary axioms}\dotfill \pageref*{elementary} \linebreak \noindent\hyperlink{sophisticated}{Sophisticated axioms}\dotfill \pageref*{sophisticated} \linebreak \noindent\hyperlink{unbiased_versions}{Unbiased versions}\dotfill \pageref*{unbiased_versions} \linebreak \noindent\hyperlink{structures_and_spaces}{Structures and spaces}\dotfill \pageref*{structures_and_spaces} \linebreak \noindent\hyperlink{generating_quasiprometrics}{Generating (quasi)-prometrics}\dotfill \pageref*{generating_quasiprometrics} \linebreak \noindent\hyperlink{morphisms}{Morphisms}\dotfill \pageref*{morphisms} \linebreak \noindent\hyperlink{subcategories}{Subcategories}\dotfill \pageref*{subcategories} \linebreak \noindent\hyperlink{proarrows}{Categorical interpretation}\dotfill \pageref*{proarrows} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions} Let $X$ be an abstract [[set]]. For purposes of this definition, let a \emph{distance function} on $X$ be a [[positive number|nonnegative]]-[[extended real number|extended]]-[[real number|real]]-valued [[binary function|binary]] [[function]] on $X$; that is, a function $d\colon X \times X \to [0,\infty]$. (For most purposes, we may assume that these distance functions are \emph{pointwise-bounded}: taking only finite values. On the other hand, for full generality in [[constructive mathematics]], we must allow the distance functions to take nonnegative extended \emph{[[upper real|upper]]} real values, although again we may assume them to be pointwise-bounded and pointwise-[[located real number|located]] for many purposes.) Let $G$ be a collection of such distance functions. \hypertarget{elementary}{}\subsubsection*{{Elementary axioms}}\label{elementary} Consider the following potential properties of $G$: \begin{enumerate}% \item \emph{Reflexivity}: For every $d \in G$ and $x \in X$, we have $d(x,x) = 0$. \item \emph{Transitivity} (a version of the [[triangle identity]]): For any $d \in G$ there exists an $e \in G$ with \begin{displaymath} d(x,z) \leq e(x,y) + e(y,z) \end{displaymath} for all $x,y,z \in X$. \item \emph{Symmetry}: For every $d \in G$, there is an $e \in G$ with $d(x,y) \leq e(y,x)$ for all $x,y \in X$. (In light of Isotony below, we may require $d(x,y) = e(y,x)$.) \item \emph{Nontriviality}: There exists a $d \in G$. (In light of Isotony below, we may require $d(x,y) = 0$ for all $x,y \in X$.) \item \emph{Filtration}: For all $d,e \in G$, there is an $f \in G$ with $d(x,y) \leq f(x,y)$ and $e(x,y) \leq f(x,y)$ for all $x,y \in X$. (In light of Isotony below, we may require $f(x,y) = \max(d(x,y), e(x,y))$ for all $x,y \in X$.) \item \emph{Isotony}: If $d \in G$ and $e$ is a distance function with $e(x,y) \leq d(x,y)$ for all $x,y \in X$, then $e \in G$. \end{enumerate} \hypertarget{sophisticated}{}\subsubsection*{{Sophisticated axioms}}\label{sophisticated} With the aid of [[abstract algebra]] (but nothing too fancy), we may view these same 6 axioms in another light, as follows. First, the [[function set]] $[0,\infty]^{X \times X}$ of all distance functions $d\colon X \times X \to [0,\infty]$ on $X$ is a [[lattice group|lattice]] $*$-[[star-monoid|monoid]] under these operations: \begin{itemize}% \item The order $\leq$ is given pointwise: \begin{displaymath} d \leq e \;\iff\; \forall(x,y\colon X),\; d(x,y) \leq e(x,y) . \end{displaymath} \item The corresponding [[join]] operation $\vee$ is also pointwise: \begin{displaymath} (d \vee e)(x,y) \coloneqq \max(d(x,y), e(x,y)) . \end{displaymath} \item The corresponding [[bottom element]] is the [[zero function]] $0$: \begin{displaymath} 0(x,y) \coloneqq 0 . \end{displaymath} \item The monoid operation $\circ$ is defined so: \begin{displaymath} (d \circ e)(x,z) \coloneqq \inf_{y \in X} (d(x,y) + e(y,z)) . \end{displaymath} \item The corresponding [[identity element]] is the infinite [[Kronecker delta]] $\delta$: \begin{displaymath} \delta(x,y) \coloneqq \begin{cases} 0 & if\; x = y,\\ \infty & if\; x \ne y. \end{cases} \end{displaymath} (This works in [[constructive mathematics]] because we are using extended \emph{[[upper real number|upper]]} reals; $\delta(x,y)$ is the [[infimum]] of the set $\{ t\colon \mathbb{R} \;|\; t = 0 \;\wedge\; x = y \}$.) \item The [[involution]] $d \mapsto d^{\op}$ is defined so: \begin{displaymath} d^{\op}(x,y) \coloneqq d(y,x) . \end{displaymath} \end{itemize} Then the same 6 axioms may be expressed as follows: \begin{enumerate}% \item Reflexivity: For every $d \in G$, we have $d \leq \delta$. \item Transitivity: For every $d \in G$, there exists $e \in G$ such that $d \leq e \circ e$. \item Symmetry: For every $d \in G$, there exists $e \in G$ such that $d \leq e^{\op}$. (In light of Isotony, we may require $d = e^op$; in other words, $d^op \in G$.) \item Nontriviality: There exists a $d \in G$. (In light of Isotony, we may require $d = 0$.) \item Filtration: For all $d,e \in G$, there is an $f \in G$ with $d \leq f$ and $e \leq f$. (In light of Isotony, we may require $f = d \vee e$.) \item Isotony: If $d \in G$ and $e$ is a distance function with $e \leq d$, then $e \in G$. \end{enumerate} It is possible to generate these even more systematically by specifying a [[proarrow equipment]] and considering the (possibly symmetric) [[pro-object|pro]]-[[monads]] in it; see the \hyperlink{proarrows}{Categorial interpretation} below. \hypertarget{unbiased_versions}{}\subsubsection*{{Unbiased versions}}\label{unbiased_versions} Reflexivity and Transitivity are a binary--nullary pair whose [[unbiased]] combination is as follows: \begin{itemize}% \item \hyperlink{elementary}{Elementary} version: For every [[natural number]] $n = 0,1,2,\ldots$ and every $d \in G$, there exists $e \in G$ with \begin{displaymath} d(x_0,x_n) \leq \sum_{i \lt n} e(x_i,x_{i+}) \end{displaymath} for each [[list]] $x_0,\ldots,x_n \in X$. \item \hyperlink{sophisticated}{Sophisticated} version: For every [[natural number]] $n = 0,1,2,\ldots$ and every $d \in G$, there exists $e \in G$ such that $d \leq e^{\circ n}$. \end{itemize} We can combine these with Symmetry by generalizing $n$ from a natural number to a [[list]] $\epsilon$ of [[bits]] and replacing $e(x_i,x_{i+})$ with $e(x_{i+},x_i)$ (that is, replacing $e$ with $e^{\op}$ in the $i$th position) when $\epsilon_i = 1$. Similarly, Nontriviality and Filtration are a binary--nullary pair whose unbiased version states (in light of Isotony) closure under finitary joins; but this is properly discussed at [[ideal]]. \hypertarget{structures_and_spaces}{}\subsubsection*{{Structures and spaces}}\label{structures_and_spaces} A collection $G$ of distance functions that satisfies all of (1--6) is a \textbf{prometric}; if we drop (3), then we still have a \textbf{quasi-prometric}. If we are working with quasi-prometrics generally, then one that happens to satisfy (3) is called \textbf{symmetric}; in other words, a prometric is precisely a symmetric quasi-prometric. Finally, a \textbf{prometric space} is a set equipped with a prometric, and likewise a \textbf{quasi-prometric space} is a set equipped with a quasi-prometric. \hypertarget{generating_quasiprometrics}{}\subsubsection*{{Generating (quasi)-prometrics}}\label{generating_quasiprometrics} We sometimes wish to consider collections of distance functions that generate (quasi)-prometrics. In the following table, a collection $G$ satisfying the conditions listed on the left has the name on the right: \begin{tabular}{l|l} Conditions&Name\\ \hline 1,2&\textbf{pre-quasi-prometric}\\ 1,2,3&\textbf{pre-prometric}\\ 4,5&[[ideal base]]\\ 1,2,4,5&quasi-prometric \textbf{[[base]]}\\ 1,2,3,4,5&prometric \textbf{base}\\ 4,5,6&[[ideal]]\\ 1,2,4,5,6&quasi-prometric\\ 1,2,3,4,5,6&prometric\\ \end{tabular} Here we always use the original (either \hyperlink{elementary}{elementary} or \hyperlink{sophisticated}{sophisticated}) formulation of (3--5); one might consider what it means if these satisfy the stronger versions rewritten in light of Isotony, but it's getting a bit far along into [[centipede mathematics]] to actually give these things names. A (quasi)-prometric base is precisely an ideal base whose generated ideal is a (quasi)-prometric. We may also speak of a (quasi)-prometric \textbf{subbase} as an [[ideal subbase]] (that is, an arbitrary collection) whose generated ideal base is a (quasi)-prometric base, or equivalently whose generated ideal is a (quasi)-prometric. A pre-(quasi)-prometric is always a (quasi)-prometric subbase, but not conversely; but thinking too hard about subbases risks more centipedes. A pre-quasi-prometric is \textbf{symmetric} if its generated quasi-prometric is symmetric (hence a prometric); a quasi-prometric base is symmetric iff it is a prometric base (and the analogous result holds for subbases), but a symmetric pre-quasi-prometric need not be a pre-prometric. Some authors may require a strong version of Symmetry in which the distance functions $d$ are all individually required to be [[symmetric binary function|symmetric]]; that is, $d(x,y) = d(y,x)$ (or simply $d = d^{\op}$). Every prometric has a base with this property; in particular, if $G$ is a prometric, then \begin{displaymath} \{ d \in G \;|\; d = d^{\op} \} \end{displaymath} is a prometric base that generates $G$, and it is this base that some authors may refer to as the prometric itself. (I write `some authors may', but there are few authors on this subject; still, it's the sort of thing that somebody might do.) A pre-quasi-prometric $G$ is \textbf{pointwise-bounded} if every $d \in G$ is pointwise-bounded; that is, $d(x,y) \lt \infty$ for every $x,y \in X$ (or simply $d \lt \infty$). If $G$ is a (quasi)-prometric, then \begin{displaymath} G_b \coloneqq \{ d \in G \;|\; d \lt \infty \} \end{displaymath} is a (quasi)-prometric that is very similar to $G$ (in particular, they are [[uniformly equivalent]]), and some authors may prefer to work with $G_b$. Indeed, when pointwise boundedness is \emph{not} required, some authors may call the structure \textbf{extended}, as an instance of the [[red herring principle]]. However, requiring pointwise boundedness interferes with the more sophisticated approaches to (quasi)-prometrics; in particular, $\delta$ is not pointwise-bounded. (It is possible that prometric spaces would work better with a closure condition that would make $G_b$ generate $G$.) Most definitions are no more complicated when phrased in terms of (quasi)-prometric bases or even pre-(quasi)-prometrics, and some constructions give one of these more naturally than the generated (quasi)-prometric. When working in [[predicative mathematics]], it is preferable to work exclusively with (quasi)-prometric bases, as the generated (quasi)-prometric will typically be a [[proper class]]. However, it is ultimately the generated (quasi)-prometric (even if referred to only obliquely) that matters. Any (quasi)-[[gauge space|gauge]] is a (quasi)-prometric base; similarly, given any (quasi)-[[pseudometric]] $d$, the [[singleton subset|singleton]] $\{d\}$ is a (quasi)-prometric base (and its generated (quasi)-gauge in turn generates the same (quasi)-prometric). One might call a (quasi)-prometric (space) \emph{simple} if it is generated in this way by a (quasi)-pseudometric; this term is used analogously in the theory of [[syntopogenous spaces]], but the term \emph{(quasi)-pseudometrizable} is more likely to be understood. (See \hyperlink{subcategories}{Subcategories} below.) \hypertarget{morphisms}{}\subsection*{{Morphisms}}\label{morphisms} A \textbf{[[short map]]} between (quasi)-prometric spaces $X$ and $Y$ is a function $f:X\to Y$ such that for every $d\in G_Y$, we have $d\circ (f\times f) \in G_X$. We write $ProMet$ for the category of prometric spaces and short maps, and similarly $QProMet$ for the category of quasi-prometric spaces and short maps. If the (quasi)-prometrics of $X$ and $Y$ are presented by bases, then this is equivalent to saying that for any basic distance function $d$ on $Y$, there is a basic $e$ on $X$ such that $d(f(x),f(x'))\le e(x,x')$ for all $x,x'\in X$. Thus, for (quasi)-pseudometric spaces and (quasi)-gauge spaces considered as (quasi)-prometric spaces, this reduces to the usual notion of short map (i.e., distance-decreasing map). Hence the category $Gau$ of gauge spaces and short maps is included as a full subcategory of $ProMet$, and similarly with $QGau$ in $QProMet$. \hypertarget{subcategories}{}\subsection*{{Subcategories}}\label{subcategories} Since $Gau$ includes the categories of metric spaces and uniform spaces (disjointly), so does $ProMet$. Likewise, since $QGau$ includes the category of topological spaces (disjointly from metric and uniform spaces), so does $QProMet$. There is also another embedding of $Unif$ into $ProMet$, however, which is notably simpler than its embedding into $Gau$ (both easier to state and not using [[dependent choice]]). Given a [[uniform space]] $X$, we define for each [[entourage]] $U\subseteq X\times X$ a distance function \begin{displaymath} d_U(x,y) = \begin{cases} 0 & (x,y)\in U\\ \infty & (x,y)\notin U. \end{cases} \end{displaymath} (Again, this may be defined constructively as an infimum.) The collection of such $d_U$ is a base for a prometric on $X$. The short maps between such prometric spaces are precisely the uniformly continuous ones, so this defines another embedding of $Unif$ into $ProMet$. The full image of this embedding consists precisely of those prometric spaces generated by a base of $\{0,\infty\}$-valued functions. Note that replacing $\infty$ by any positive real number also defines an embedding of $Unif$ into $ProMet$, but a yet different one. Conversely, every prometric induces a uniformity, where the entourages are the sets \begin{displaymath} U_{d,\epsilon} = \{ (x,y) \;|\; d(x,y) \lt \epsilon \} . \end{displaymath} In this way every short map induces a uniformly continuous map as well. This operation is compatible with the above inclusions of $Unif$, as well as with the inclusion via $Gau$. \hypertarget{proarrows}{}\subsection*{{Categorical interpretation}}\label{proarrows} As observed by Lawvere, an extended quasi-pseudo-metric space is a [[category enriched]] over the [[monoidal category]] $([0,\infty],\geq,+,0)$. In other words, it is a monoid (or [[monad]]) in the [[bicategory]] $[0,\infty] Mat$ of [[matrices]] with values in this monoidal category. Analogously, an extended quasi-prometric space is a monad in the bicategory $Pro [0,\infty] Mat$ whose hom-categories are the categories of [[pro-object]]s in the hom-categories of $[0,\infty] Mat$. Note that if $Rel = \{0,1\} Mat$ denotes the bicategory of [[relation]]s in $Set$, then a monad in $Rel$ is a [[preorder]], while a monad in $Pro Rel$ is a quasi-[[uniform space]]. In all these cases, in order to recover the correct notion of morphism abstractly, we must consider monads in a [[double category]] or [[equipment]] rather than merely a bicategory. [[!include generalized uniform structures - table]] \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item Maria Manuel Clementino, Dirk Hofmann, and Walter Tholen. ``One Setting for All: Metric, Topology, Uniformity, Approach Structure.'' (\href{http://www.math.yorku.ca/~tholen/ProMat_V_.pdf}{pdf}) \end{itemize} [[!redirects prometric]] [[!redirects prometrics]] [[!redirects prometric space]] [[!redirects prometric spaces]] [[!redirects quasiprometric]] [[!redirects quasiprometrics]] [[!redirects quasiprometric space]] [[!redirects quasiprometric spaces]] \end{document}