%\input preamble_mrp.tex

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\DeclareMathAlphabet\mathbb{U}{msb}{m}{n}
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\usepackage[matrix, arrow,all,cmtip,color]{xy}
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\newcommand{\bi}{\begin{itemize}}
\newcommand{\ei}{\end{itemize}}
\newcommand{\bd}{\begin{description}}
\newcommand{\ed}{\end{description}}


\newcommand{\bee}{\begin{enumerate}}
\newcommand{\eee}{\end{enumerate}}


\theoremstyle{plain}
\newtheorem{conjecture}{Conjecture}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}

\newtheorem{prop}{Proposition}
\newtheorem{claim}{Claim}


\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem{question}{Question}
\newtheorem{rem}{Remark}
\newtheorem{remark}{Remark}
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\def\bqqq{\begin{quote}}
\def\eqqq{\end{quote}}

\newcommand*{\longhookrightarrow}{\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}}
\def\lhra{\longhookrightarrow}
\def\lra{\longrightarrow}
\def\ra{\rightarrow}

\def\rtt{\,\rightthreetimes\,}



\def\xra{\xrightarrow}

\def\ZZ{\Bbb Z}
\def\CC{{\mathcal C}}
\def\wra{{\lra}}
\def\xra{\xrightarrow}


\def\rrt#1#2#3#4#5#6{\xymatrix{ {#1} \ar[r]^{} \ar@{->}[d]_{#2} & {#4} \ar[d]^{#5} \\ {#3}  \ar[r] \ar@{-->}[ur]^{}& {#6} }}

\def\union{\cup}

\def\NN{\Bbb N}
\def\RR{\Bbb R}

\def\card{\,{\mathrm{card}\,}}
\def\uU{\mathcal U}
\def\XxX{X\times X}
\def\XxK{X\times K}
\def\UxV{U\times V}
\def\EA{\exists\forall}
\def\AE{\forall\exists}

%%%%%%%%%%%%%%%%%%%5

\def\salpha{s_{<\alpha}}

\def\Sets{\mathrm{Sets}}
\def\sSets{\mathrm{sSets}}
\def\ssSets#1{\mathrm{s_{<#1}Sets}}

\def\Topp{\mathrm{Top}}
\def\sTop{\mathrm{sTop}}

\def\FF{\mathcal F}
\def\LL{\mathcal L}

\def\NN{\mathbb N}

\def\ssTop#1{\mathrm{s_{<\!#1}Top}}
\def\ssaTop{\ssTop\alpha}
\def\Metric{\mathrm{Mtr}}
\def\MUniform{\mathcal{MU}}
\def\MLipschitz{\mathcal{ML}}
\def\MLipschitzOne{\mathcal{ML}_1}
\def\MLips{\MLipschitz}
\def\MLipsOne{\MLipschitzOne}
\def\MDadd{{\mathcal M}_D}
\def\qLips{{\mathcal{ML^q}}}
\def\Ordfiniop{\mathrm{Ord_{<\omega}^{op}}}
\def\Ordwop{\mathrm{Ord_{\leq\omega}^{op}}}
\def\Ord#1{\mathrm{Ord}_{<#1}}

%\def\Filt{\text{\ethmath{wA}}\!\mathit{ilt}}
%\def\sFilt{s\text{\ethmath{wA}}\!\mathit{ilt}}

\def\Filt{{\ethi\ethmath{wA}}\!\mathit{ilt}}
\def\sFilt{s{{\ethi\ethmath{wA}}}\!\mathit{ilt}}

\def\sFilth{\mathit{sFilt}}



%\def\FFilt{{\ethi\ethmath{qa}}\!\text{\it ilt}}
%\def\sFFilt{s{\ethi\ethmath{qa}}\!\text{\it ilt}}
%\def\sFFilth{\mathit{sFFilt}}
\def\FFilt{{\ethi\ethmath{wE}}\!\text{\it ilt}}
\def\sFFilt{s{\ethi\ethmath{wE}}\!\text{\it ilt}}
\def\sFFilth{\mathit{sFFilt}}


\def\Fmilt{{\ethi\ethmath{qa}}\!\text{\it ilt}}
\def\sFmilt{s{\ethi\ethmath{qa}}\!\text{\it ilt}}




\def\Ee{\mathrm{E}}



\def\mD{m_{ D}}

\def\Ed{\mathrm{E_\Delta}}



\def\iemb{{\ethi{\ethmath{nI}}}}


\def\iemba{{\ethi{\ethmath{nA}}}}
\def\iembd{{\ethi{\ethmath{nE}}}}



\def\ttt{{\ethi{\ethmath{pa}}}}

\def\tttt{{\ethi{\ethmath{pA}}}}

\def\mU{{\ethi{\ethmath{mi}}}}


\def\mmU{{\ethi{\ethmath{mA}}}}

\def\mD{m_{ D}}
\def\mL{m_{\mathcal L}}
\def\mLOne{m_{\mathcal L_1}}

\def\hhom{\underline{Hom}}

%%%%%%%%%%%%%%%%

%%%
%%%\title[Expressing the statement of Feit-Thomson theorem
%%%with diagrams in the category of finite groups
%%%\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\textbf{\tiny To}  {\textsf{\bf Grigori Mints Z"L}} \tiny In memoriam}
%%%]{Expressing the statement of Feit-Thomson theorem
%%%with diagrams in the category of finite groups
%%%\\{\tiny.}
%%%\\
%%%{{{\tiny To}  {\textsf{Grigori Mints Z"L}} \tiny In memoriam}}
%%%}
%%%\author[Misha Gavrilovich]{Misha Gavrilovich\ \ \ \ \ \
%%%{{{\tiny }  {\text{July 2016}}}}}
%%%\date{2013}
%%%
%\extraline{\copyright\ 2014 2015 Misha Gavrilovich {	tt mi\!\!\!ishap\!\!\!p@sd\!\!\!df.org http://mishap.sdf.org/mints-lifting-property-as-negation}}
%\classno{97A80, 97B70}
%\journal{ mishap.sdf.org/mints \hfill
%\hfill ISSN 2053-1451}
%\volume{5}
%\parskip 5pt
%\setcounter{page}{23}

%\maketitle

%\text{{\ethi evgenii shurygin nI gA gI gE ga gi ge .cE ge  ne }}
%\setlength{\epigraphwidth}{0.5\textwidth}
%\epigraph{\normalsize There's no point in being grown up if you can't be childish sometimes
%}{ {\emph{ %fixme: a better epigraph is needed.
%}}
%}


\begin{document} 
\selectlanguage{english}\catcode`\_=8\catcode`\^=7
\date{}%9 May 2017}
\title{
%%%%n 
Topological and metric spaces are full subcategories of the category of simplicial objects of the category of filters.
%%....sFilters...
%%Metric spaces as a subcategory of simplicial topological spaces
%%%%A diagram chasing formalisation of elementary topological properties
%The unreasonable power of the lifting property in elementary mathematics
}
\author{
%%%%n 
%an early draft of a research proposal\\ 
notes by misha gavrilovich\thanks{%very early draft with known misprints; 
Comments welcome.  
 $\tt{mi\!\!\!ishap\!\!\!p@sd\!\!\!df.org}$. 
I thank Dmitry Krachun, Sergei Ivanov and Vladimir Sosnilo for discussions. 
% This draft is essentially a part of a more verbose draft on the expressive power of the lifting property available at http://mishap.sdf.org/expressive-power-of-the-lifting-property.pdf.  
}\\
%in memoriam: evgenii shurygin
}
\maketitle
%%epigraph:
%\setlength{\epigraphwidth}{0.8\textwidth}
%%%\epigraph{%In the following sections we bring up 
%%%\small.\hskip-18pt instances of human and animal behavior
%%%%%%%   which are, on one hand,
%%%[...]  miraculously complicated, 
%%%%%%%on the other hand 
%%%%%%%[...] they have
%%%[of] little, if any, pragmatic (survival/reproduction) value, %.
%%%%%%%   From this we conclude [...]
%%%%%%%   %that since the corresponding features of ergobrains were not the primarily targets
%%%%%%%   %pecifically selected for by the evolution, 
%%%%%%[...]   they 
%%%are due to internal constraints on
%%% possible architectures of unknown to us functional "mental structures".
%%%%%%}{
%%%%%%   Gromov, Ergobrain
%%%}
%%%
\begin{abstract}

%\text{{\ethi evgenii shurygin nI gA gI gE ga gi ge .cE ge  ne }}

We observe that the category of topological space, uniform spaces, and simplicial sets 
are all, 
in a natural way, 
full subcategories of the same larger category, namely the simplicial category of filters;
coarse space of large scale metric geometry are also similicial objects of a category of filters with different morphisms. 
This is, moreover, implicit in the definitions of a topological, uniform, and coarse space.

We use these embeddings  to rewrite the notions of completeness, precompactness, compactness, Cauchy sequence, and equicontinuity
in the language of category theory, which we hope might be of use in formalisation of mathematics and tame topology. 
We formulate some arising open questions. 
%Our observation suggests a number of open questions and we formulate some.


%We rewrite  several notions of elementary topology in the first two chapters (Bourbaki, General Topology), notably
%compactness, completeness, equicontinuity, 
%in the language of category theory using embeddings of topological and metric spaces into the category of simplicial objects 
%of the category of filters. This approach suggests a number of open questions, which we formulate, largely for the author's own use. 
%
%This is an early draft of a research proposal. 
%

%%%%+ abstract:
%....We show that metric spaces with uniformly continious maps embed as a full subcategory in the category of simplicial topological spaces,
%and ..also .. metric spaces with quasi-Liplistz and quasi-isomotercie is a full subcategry of ???. 
%We introduce a combinatorial formalism of category theoretic flavour which concisely expresses a number of 
%elementary topological properties, e.g. compactness, connectedness, dense image, separation axioms. 
%We hope these expressions may be use in formalisation of elementary topology and in teaching. 
%%%These expressions are based on the simplest (counter)example and in this sence are intuitive, 
%%%and, arguably, are short enough (several bytes) to be found by a brute force search
%%%and may be of use in an automatic theorem prover.
\end{abstract}

\section{
%%%% 
Introduction.
}
In this proposal we %observe %that the definitions of a topological space and a uniform space
%define two full subcategories of the category of simplicial objects in the category of filters,
define two fully faithful embeddings of the category of topological spaces and 
that of uniform metric  spaces into the category of simplicial objects of the category of filters,
and, based on this, use these two functors to reformulate several elementary  notions 
including that of being compact, precompact, complete, a Cauchy sequence, and equicontinuity. 

We formulate a number of open questions, largely for the author's own use: this proposal 
is at a very early stage and it is possible that some of the questions are easy or indeed well-known. 

If someone already knows the answers or the relevant literature, 
the author would be delighted to hear about it.     

We hope our reformulations suggest that a number of notions as defined in [Bourbaki, General Topology] 
may conveniently and concisely be expressed in the language of category theory. 
It may be worthwhile to express them this way, for two reasons: it may provide a fresh point of view
on foundations of topology (tame topology) and it may lead to a development of the language of category theory.
It is possible that this may be of use in formalisation of foundations of topology.



%It is quite possible we are unaware of some relevant literature. 


\section{Main constructions and open questions.}

\subsection{The category of simplicial filters.}
We say a topological space is {\em filtered} iff 
\bi\item[($F_I$)] any superset of a non-empty open set is open.  
\ei 
We say a subset of a topological space is {\em big} iff it is non-empty and open. 


A {\em filter} is a filtered topological space $X$ which is not discrete.
In a filter %($F_{ii}$) 
the intersection of two big, i.e. non-empty open, subsets is big, i.e. non-empty.
Indeed, for any two disjoint open non-empty subsets $U$ and $V$, an arbitrary subset $X$
is the intersection $X=(U\cap V)\cup X=(U\cup X)\cap (V\cup X)$ of two open non-empty subsets 
$U\cup X$ and $V\cup X$.





Let $\Filt$ be the full subcategory of the category of topological spaces 
whose objects are filtered topological spaces.
Let $\FFilt$ be the category with the same objects but maps considered up to being equal almost everywhere, i.e.
two continuous maps between filtered topological spaces are considered equal in $\FFilt$ iff they coincide 
on a big subset of the source. 

Let $\Fmilt$ be the category whose objects are filters but morphisms defined differently:
an $\Fmilt$-morhpism is  a map of underlying sets such that the image of a small subset
is necessarily small.   

%${\ethi\ethmath{qa}}$ $s{\ethi\ethmath{qa}}$ ${\ethi\ethmath{wa}}$ $s{\ethi\ethmath{wA}}$ qq ${\ethi\ethmath{wE}}$ $s{\ethi\ethmath{qA}}$




 
The categories $\Filt$ and $\FFilt$ both have all small limits and colimits and a non-commutative tensor product [Blass,Thm.7]. Limits and colimits 
 are set-wise the same as in $Sets$ 
and the topology is defined as the finest/coarsest filtered topology such that the necessary maps 
are continuous.  
Let $\sFilt$ be the category of simplicial objects in the category $\Filt$
of filtered topological spaces, i.e. 
$\sFilt=Func(\Ord{\omega}^{op},\Filt)$
where $\Ord{\omega}$ denotes the category of  categories corresponding to finite linear orders 
$$\bullet_1\lra .. \lra \bullet_n,\ 0\leq n <\omega.$$ 


There are two natural functors $\Filt\lra \sFilt$:
\bi\item[]
$\iemb:F \longmapsto (F,F,F,...), \text{ identity maps }$
\item[] $E:F \longmapsto (F,F\times F, F\times F\times F, ....)$,
face and degeneracy maps 
 are coordinate maps $F^n\lra F^m, (x_1,...,x_n)\mapsto (x_{i_1},...,x_{i_m})$ where $1\leq i_1\leq i_2\leq ...\leq i_m\leq n$.
\ei

The same considerations apply to $\FFilt$ and $\sFFilt$. 

There are two natural inclusions $\iemba:\Sets\lra \Filt$: a set $S$ goes to the filter on $S$ with the unique big subset $S$, 
and 
$\iembd:\Sets\lra \Filt$: 
a set $S$ goes to the filter on $S$ where all non-empty subsets are big. 
This gives three fully faithful embeddings $s\iemba:\sSets\lra \sFilt$, $s\iemba':\sSets\lra \sFFilt$, and 
$s\iembd:\sSets\lra \sFilt$.


\subsection{Topological and metric spaces as simplicial filters} 

\subsubsection{Topological and uniform spaces}
Topological and uniform spaces are defined [Bourbaki, Ch1., Ch.2] as systems of neighbourhood filters satisfying 
certain compatibility conditions, %in fact objects of $\sFilt$,
and lead us to define two fully faithful functors $\ttt:Top\lra \sFilt$, $\mU:\MUniform\lra\sFilt$,
and in fact also  
 two fully faithful functors $\tttt:Top\lra \sFFilt$, $\mmU:\MUniform\lra\sFFilt$.

In fact, everything we say below about $\Filt$ and $\sFilt$ holds also for $\FFilt$ and $\sFFilt$, i.e. 
when maps are considered up to being equal almost everywhere. 

In Appendix~B we show how to ``read off'' the latter embedding  from the definition of uniform structures in 
[Bourbaki, Chapter 2]. 




 
For a topological space $X$, let $\ttt(X)$ denote the following object in $\sFilt$.
%Let $t:Top\lra \sFilt$ be the following functor. A space $X$ maps to the object
   $$\ttt:X\longmapsto 
(|X|,|X|\times |X|, |X|\times |X| \times |X|, ...)$$
with face and degeneracy maps being the coordinate maps 
$$|X|^n\lra |X|^m, (x_1,...,x_n)\mapsto (x_{i_1},...,x_{i_m})$$ where $1\leq i_1\leq i_2\leq ...\leq i_m\leq n$.

 
A subset $U=\{(x_1,...,x_n):(x_1,...,x_n)\in U\}$ of $|X|^n$ is {\em big} iff the following formula holds:
\bi
\item[]  $\forall x_1\in X \exists U_{x_1}\ni x_1 \text{ where }U_{x_1}\text{ is a neighbourhood of } x_1 \\ 
\forall x_2\in U_{x_1} \exists U_{x_2} \ni x_2 \text{ where }U_{x_2} \text{ is a neighbourhood of } x_2 \\
....\\
  \forall x_n\in U_{x_{n-1}} \exists  U_{x_n} \ni x_n \text{ where }U_{x_n} \text{ is a neighbourhood of } x_n \\ 
 \text{ \ } (x_1,x_2,...,x_n) \in U  $
\ei

Note that 
any big subset of $|X|^n$ contains the diagonal, in particular 
the topology on $|X|$ is always antidiscrete.
Topology on $X$ is discrete iff the diagonal in $|X|^n,n\geq 2$ is open, equivalently a subset of $|X|^n$ is big iff it contains the diagonal.

For $X$ finite, % the filter on 
a subset of $|X|^n,n\geq 2$ is big iff it contains all the non-strictly decreasing sequences in the specialisation preorder, i.e. 
all the sequences  $(x_1,x_2,...,x_n)$ 
such that $x_i\in cl_X(x_{i+1})$, $1\leq i <n$.
%For $X=\{0\lra 1 \}$ the Sierpienki space consisting of the open point $0$ and closed point $1$, the smallest open subset 
%of $|X|^n=\{0,1\}^n$ containing the diagonal consists of non-increaing sequences $11...11, 11...10, 11...100, ... 10...0, 00...0$, i.e.
%a sequence of 1's followed by a sequence of 0's.

Note that the topology on $|X|^n$ is the coarsest filter such that the maps 
$|X|^n \lra |X|^2, (x_1,...,x_n)\mapsto (x_i,x_{i+1})$, $1\leq i< n$ 
are continuous.

For a metric space $M$, let $\mU(M)$ denote the following object in $\sFilt$.
$$\mU:M\longmapsto (|M|,|M|\times |M|, |M|\times |M| \times |M|, ...)$$ 
where a subset of $|M|^n$ is {\em big} iff it contains an $\varepsilon$-neighbourhood of the diagonal $\{ (x,..,x):x \in M\}$.
Face and degeneracy maps are coordinate maps 
$$|X|^n\lra |X|^m, (x_1,...,x_n)\mapsto (x_{i_1},...,x_{i_m})$$ where $1\leq i_1\leq i_2\leq ...\leq i_m\leq n$.

Note that, as before,  
any big subset of $|M|^n$ contains the diagonal, in particular 
the topology on $|M|$ is always antidiscrete.  
The diagonal in $|M|^n,n\geq 2$ is open, %equivalently a subset of $|X|^n$ is big iff it contains the diagonal, 
iff the metric space $M$ is discrete.

Note that the topology on $|M|^n$ is the coarsest filter such that the maps 
$|X|^n \lra |X|^2, (x_1,...,x_n)\mapsto (x_i,x_{j})$, $1\leq i<j\leq n$
are continuous.

Let $\MUniform$ denote the category of uniform spaces [Bourbaki, II\S1.1]. In a similar way 
$\mU(M)$ is defined also for $M$ a uniform space. 


Note that permutations of coordinates act on $\mU(M)$. 

\begin{remark} The intuition behind these definitions is as follows.
[Bourbaki,Introduction] writes: `a topological structure now enables us to give precise meaning to the
phrase ``such and such a property holds for all points sufficiently near a":
by definition this means that the set of points which have this property is
a neighbourhood of a for the topological structure in question.' 
The notion of a {\em big subset} enables to concisely express the
phrase ``such and such a property holds provided a point $a_n$ is sufficiently near $a_{n-1}$
which in turn is sufficiently near $a_{n-1}$, which in turn is sufficently near ...,
which is turn is sufficiently near $a_1$". 
By definition this means that the set of tuples $(a_1,\ldots,a_n)$ of points which have this property is
big for the topological structure in question.

Further, [Bourbaki, Introduction] writes: `As we have already said, a topological structure on a set enables one
to give an exact meaning to the phrase ``whenever $x$ is sufficiently
near $a$, $x$ has the property $P\{x\}$". But, apart from the situation
in which a ``distance" has been defined, it is not clear what meaning
ought to be given to the phrase ``every pair of points $x,y$ which are sufficiently near each other has 
the property $P \{x, y\}$", since a priori
we have no means of comparing the neighbourhoods of two different
points. Now the notion of a pair of points near to each other arises fre-
quently in classical analysis (for example, in propositions which involve
uniform continuity). It is therefore important that we should be able
to give a precise meaning to this notion in full generality, and we are
thus led to define structures which are richer than topological structures,
namely {\em uniform structures}.'

The notion of a {\em big subset} in a uniform space enables to concisely express the
phrase ``such and such a property holds provided points $a_1,\ldots,a_n$ are sufficiently near to each other". 
By definition this means that the set of tuples $(a_1,\ldots,a_n)$ of points which have this property is
big for the uniform structure in question.




\end{remark}



\begin{claim} $\ttt:Top\lra \sFilt$, and $\mU:\MUniform\lra\sFilt$, 
and  
$\tttt:Top\lra \sFFilt$, and $\mmU:\MUniform\lra\sFFilt$, 
are fully faithful functors.
\end{claim}
\proof The verification is straightforward and we only consider $\ttt$. 
The formula is positive and therefore a superset of a big subset is big.
The intersection of two neighbourhoods is a neighbourhood [Bourbaki,I\S1.2,Ax.$(V_{II})$] and this carries though the quantifiers.
Finally, each neighbourhood of a point contains the point, and this implies that big subsets necessarily contain the diagonal
and thus form a filter. To see continuity of a degeneracy map $(x_1,...,x_n)\longmapsto (x_1,..,x_i,x_{i+1},...,x_n)$, 
pick the same neighbourhood twice, $U_{x_i}=U_{x_{i-1}}$ or  $U_{x_1}=X$ if $i-1<1$; 
this uses that an open subset is a neighbourhood of each of its points [Bourbaki,I\S1.2,Ax.$(V_{IV})$] 
and that $X$ itself is open [Bourbaki,I\S1.2,Ax.$(V_{I})$]. 
To check continuity of a face map 
$(x_1,...,x_n)\longmapsto (x_1,..,x_i,x_i,x_{i+1},...,x_n)$, pick $x_{i+1}=x_i$; this uses that a point is contained 
in any neighbourhood of itself [Bourbaki,I\S1.2,Ax.$(V_{III})$]. 
 The functor is faithful because the morphism on $X^1$ uniquely determines morphisms on all the other Cartesian powers. 
 A function $f:X\lra Y$ is continuous iff for each point $x\in X$ and each neighbourhood $V_y$ of $y=f(x)$ there is a neighbourhood $U_x$ 
such that $V_y\subset f(U_x)$. This is implied by the fact that the preimage of a big set $\{y\}\times V_y \cup \cup\limits_{y'\neq y} \{y'\}\times Y$ 
contains $\{x\}\times U_x$ for some $U_x$, as it is big. 
 \qed



For a metric space $M$, the functor $\mU(M): \Ord{\omega}^{op} \lra \sFilt$ uniquely factors via $FinSets^{op}$:
$\Ord{\omega}^{op} \xra{id} FinSets^{op} \lra \sFilt$. And in fact it might be better to view it this way:


\begin{claim} To give a uniform space is the same as to give a functor 
$$M:FinSets^{op} \lra \Filt$$
such that \bi
\item[] $|M(S)|=X^S=Hom_{Sets}(S,X)$, for any finite set $S$ and some fixed set $X$ 
\item[] the only non-empty open subset of $M(\{\bullet\})$ is $|M(\{\bullet\}|$ itself

\item[] the filter on $M(S)$ is the coarsest filter such that the  induced maps 
$$M(S)\xra{M(\{s_i,s_{i+1}\}\lra S)}  M(\{s_i,s_{i+1}\})$$ 
     are continuous for the inclusion maps $\{s_i,s_{i+1}\}\lra S$, $1\leq i\leq \card(S)$,
    for some, equiv. each, enumeration $S=\{s_1,...,s_{\card(S)}\}$ 


\ei
Equivalently, 
 to give a uniform space is the same as to give a similicial object 
$$M:\Ord{\omega}^{op} \lra \Filt$$ in the category $\Filt$
such that \bi
\item[] $|M_n|=Hom_{Sets}(\{0,..,n\},X)$, for  some fixed set $X$ and any $n\geq 0$
\item[] in $M_0$, the only non-empty open subset is  $|M_0|$ itself
\item[] it extends to a functor $M:FinSets^{op} \lra \Filt$
\item[] the filter on $M_n$ is the coarses filter such that the  induced maps 
$$M_n\xra{pr_{x_i,x_{i+1}}}  M_1$$ 
     are continous.
\ei


\end{claim}

\proof It is straightforward to verify that the construction above gives such a functor. 
Conversely, such a functor defines a filter on $X\times X$, for some set $X$,
and it is straightforward to verify this filter determines a uniform structure on $X$
using the  axioms [Bourbaki,II \S1.1,Ax.$(U_i)-(U_{III})$].
 \qed

\subsubsection{Coarse spaces}

Replacing in Claim~2 the category $\Filt$ by $\Fmilt$ leads to the definition of a coarse structure ([Bunke,Engel], Def.2.3).

Let us start with a construction of a coarse structure associated to a metric space and some remarks. 
Let $X$ be a metric space. Call a subset $U$ of $X^n$ {\em small} iff the diameters of tuples in $U$ are uniformly bounded,
i.e. there is a $d=d(U)$ such that for each $(u_1,...,u_n)\in U$, $dist(u_i,u_j)\leq d$ for each $1\leq i,j\leq n$;
this defines a filter on $X^n$. 
Note that coordinate maps $X^n\lra X^m$ have the property that the image of a small subset is necessarily small.
Hence this construction defines a functor ${\mathcal X}:FinSets^{op} \lra \Fmilt$. 
A natural transormation $\mathcal X\lra \mathcal Y$ of functors associated with metric spaces $X$ and $Y$, resp.,
corresponds to a map of metric spaces $f:X\lra Y$ such that
for each $d>0$ there is $D>0$ such that $$dist(f(x'),f(x''))<D\text{ whenever }dist(x',x'')<d, x',x'' \in X.$$ 
For $X$ quasigeodesic, this is the class of large scale Lipschitz maps. 


Recall the definition of a coarse structure ([Bunke,Engel],Def.2.3): 

\begin{defn}%Definition 2.3. 
A {\em coarse structure} on a set  $X$ is a collection of subsets of $X \times X$ which is closed under
taking finite unions, composition, inverses and subsets, and which contains the diagonal $\{(x,x):x\in X\}$
of $X$.
\end{defn}


\begin{claim} To give a coarse  structure  is the same as to give a functor 
$$M:FinSets^{op} \lra \Fmilt$$
such that \bi
\item[] $|M(S)|=X^S=Hom_{Sets}(S,X)$, for any finite set $S$ and some fixed set $X$ 
\item[] in $M(\{\bullet\})$, the subset  $|M(\{\bullet\}|$ is small 

\item[] the filter on $M(S)$ is the finest filter such that the  induced maps 
$$M(S)\xra{M(\{s_i,s_{i+1}\}\lra S)}  M(\{s_i,s_{i+1}\})$$ 
     are well-defined $\Fmilt$-morphisms for the inclusion maps $\{s_i,s_{i+1}\}\lra S$, $1\leq i\leq \card(S)$,
    for some, equiv. each, enumeration $S=\{s_1,...,s_{\card(S)}\}$ 
\ei
Equivalently, 
 to give a coarse  structure  is the same as to give a similicial object 
$$M:\Ord{\omega}^{op} \lra \Fmilt$$ in the category $\Fmilt$
such that \bi
\item[] $|M_n|=Hom_{Sets}(\{0,..,n\},X)$, for  some fixed set $X$ and any $n\geq 0$
\item[] in $M_0$, the subset  $|M_0|$ is small 
\item[] it extends to a functor $M:FinSets^{op} \lra \Fmilt$
\item[] the filter on $M_n$ is the finest filter such that the  induced maps 
$$M_n\xra{pr_{x_i,x_{i+1}}}  M_1$$ 
     are well-defined $\Fmilt$-morphisms.
\ei


\end{claim}

{\bf Proof} is a verification. We only show that such a functor gives rise to a coarse structure.
$M_1=X\times X$ and thus small subsets of $M_1$ define a collection of subsets of $X\times X$. 
To check it contains the diagonal, consider the map $X=M_0\lra M_1=X\times X$; its image is small 
because $M_0$ is assumed small. To check it is closed under inverses, consider 
the involution of $M_1$ corresponding to the involution of $\{0,1\}$.
Finally, use the last item to check it is closed under compostion.
\qed


\subsubsection{Questions.}


These two embeddings immediately give rise the following questions.

\begin{question}\small\upshape\small\upshape(Category theory and homotopy)
\bee
\item Do functors $\ttt,\mU$ have adjoints? Do they preserve limits and colimits? 
\item  Characterise the full subcategories $\ttt(Top)$ and $\mU(\MUniform)$ of $\sFilt$ in terms of the ambient category $\sFilt$ or $\sFFilt$. 
\item Characterise systems of neighbourhoods such that the construction of $\ttt$ %formula above 
gives rise to a simplicial object. 
\item  Does a model structure on $Top$ extend to a model structure on $\sFilt$ or $\sFFilt$? Does $\sFilt$ or $\sFFilt$ have an interesting model structure? 
\eee
\end{question}

The following questions are vague.

Everywhere below we may talk of $\sFFilt$ instead of $\Filt$, and sometimes we omit $\sFFilt$. 

\begin{question}\small\upshape(Naive homotopy theory)
\bee
\item What is the ``right'' notion in $\sFilt$ or $\sFFilt$ of a real line interval $[0,1]$, a fibration, and path, loop, and suspension objects? 
\item Is there an interesting object of $\sFilt$ which corresponds to the path space of a topological space and which is more ``finitary''? 
Note that in $\sFilt$ topological spaces have dimension $2$ and that path space is thought of a space ``shifted''. 
Can this ``shift'' be realised in $\sFilt$ somehow, e.g.~so that the $\sFilt$-path space of a topological space have dimension $3$ ? 
\item Is there an interesting object of $\sFilt$ which corresponds to a foliation, particularly an irrational foliation? 

\eee\end{question}

\begin{question}\small\upshape(``Combinatorial'' definition of compactness and completeness)
\bee
\item A number of elementary topological properties can be defined by, in a sense, combinatorial expressions,
   by taking iterated orthogonals in $Top$ 
   of a single %of classes consisting of a single 
   morphism between finite topological spaces [Gavrilovich, Lifting Property]. Calculate %evaluate 
     these expressions in $\sFilt$ using the embedding $\ttt:Top\lra \sFilt$. Note that this would give properties of both topological spaces 
     and metric spaces.  Do they define the same properties of topological spaces? Do they provide an interesting analogy between topological spaces 
and metric spaces, e.g. compactness [Bourbaki, I\S10.2, Thm.1(d), p.101] and completeness [Bourbaki, II\S3.6, Prop.11]? 
\item The following is an example of a precise conjecture. Evaluated in $Top$, the following expression defines the class of almost(?) all proper maps 
[Gavrilovich, Lifting Property, Claim 1]:
$$ ((\{ \{o\}\lra \{o\ra c\}\}^{r})_{<5})^{lr}
$$
If evaluated in $\sFilt$, does the same expression define both the class of compact space and complete metric space, i.e. is the following true:
 \bee
\item A Hausdorff space $X$ is compact iff $$\ttt\left(X \lra \{pt\}\right) \in \left((\{\ttt\left(\{o\}\lra \{o\ra c\}\right)\}^{r})_{<5}\right)^{lr} $$ 
\item A metric space $M$ is complete iff $$\mU\left(M\lra \{pt\}\right) \in \left((\{\ttt\left( \{o\}\lra \{o\ra c\}\right)\}^{r})_{<5}\right)^{lr} $$
\eee
\eee

\end{question}



\begin{question}\small\upshape(Tame topology of Grothendieck)
\bee
\item Does this point of view shed light on tame topology of Grothendieck, 
k, i.e. a foundation of topology
``without false problems" and ``wild phenomena" ``at the very beginning"?
Is there a better construction of ``the''  tubular
neighbourhood of closed tame subspace in a tame space? I quote a specific suggestion by 
Grothendieck [Esquisse d'un Programme, translation,\S5,p.33]: 
\begin{quote}
   Among the first theorems one expects in a framework of tame topology as
I perceive it, aside from the comparison theorems, are the statements which
establish, in a suitable sense, the existence and uniqueness of ``the" tubular
neighbourhood of closed tame subspace in a tame space (say compact to
make things simpler), together with concrete ways of building it (starting
for instance from any tame map $X \lra {\Bbb R}^+$ having $Y$ as its zero set), the
description of its ``boundary" (although generally it is in no way a manifold
with boundary!) $\partial T$, which has in $T$ a neighbourhood which is isomorphic to
the product of $T$ with a segment, etc. Granted some suitable equisingularity
hypotheses, one expects that $T$ will be endowed, in an essentially unique
way, with the structure of a locally trivial fibration over $Y$, with $\partial T$ as a
subfibration.
 \end{quote}
\eee
\end{question}

\begin{question}\small\upshape(History and formalisation of mathematics) 
\bee
\item[] Early works on topology talk about topological spaces in terms of neighbourhood systems.
Could it be that they are implicitly trying to express (say, functorial) constructions in $\sFilt$, 
are implicitly using category theoretic language but describing it in words? Can this question be made precise? 

For example, it was a convention to always mean by $U(x)$ a neighbourhood of a point $x$, and that's somewhat natural 
from the point of view of our definition of $\ttt:Top\lra \sFilt$.

In Appendix~B we show how to ``read off'' our construction of $\mU:\MUniform\lra \sFilt$ from Bourbaki. Arguably, $\ttt:Top \lra \sFilt$
and the combinatorial definitions of elementary topological properties  [Gavrilovich, Lifting Property] are ``implicitly contained'' in Bourbaki. 
In what sense are these reformulations  ``contained implicitly'' there? Can this sense be made explicit?

Could these reformulations be of use in formalisation of topology and analysis, e.g. Chapter 1 and 2 of Bourbaki?  
\eee
\end{question}


\subsection{Elementary theory of topological and metric spaces
% in terms of embeddings $\ttt:Top\lra \sFilt$, $\mU:\MUniform\lra \sFilt$ 
}

We reformulate several notions from [Bourbaki, Chapter 1, 2] %, Uniform Structures].
in terms of functors $\ttt:Top\lra \sFilt$ and $ \mU:\MUniform\lra \sFilt$. 

%We reformulate several notions in terms of the embedding 
%$m_{\mathcal U}: \MUniform \lra Func(\Ordfiniop, \Topp)$.

\subsubsection{Compact and complete metric spaces}

%%A {\em filter} is a topological space $\mathcal F$ such that (i) a superset of a non-empty open set is open 
%%(ii) the intersection of two non-empty open sets is open. 
An {\em ultrafilter} is a filter such that if the union of finitely many sets is open, then one of them is; equivalently, 
each subset $A$ is either open or closed.

With a filter $\mathcal F$ on the set of points of a topological space $X$ 
associate [Bourbaki, II\S5,Example] a topological space $X\cup_\FF \{\infty\}$ such that $\FF$ is the neighbourhood filter of $\infty$:
$|X\cup_\FF \{\infty\}|=|X|\cup\{\infty\}$, and a subset is open iff it is either an open subset of $X$
or contains $\infty$ and is a union of $\{\infty\}$ and an open subset of $X$ which is also $\FF$-open.

%  associate two topologies  $\mathcal{F_1}$ and $\mathcal{F}_\infty^{discrete}}$
%  on the set $|\mathcal{F_1}|=|\mathcal F| \cup \{\infty\}$ as follows.
%  A subset of $|\mathcal F| \cup \{\infty\}$ is $\mathcal{F_1}$-open if it is either a $\mathcal F$-open subset of $|\mathcal F|$ or 
%  a union of $\{\infty\}$ and an $\mathcal F$-open subset of $|\mathcal F|$.
%  
%  A subset of $|\mathcal F| \cup \{\infty\}$ is $\mathcal{F_1}^{discrete}$-open if it is either a subset of $|\mathcal F|$ or 
%  a union of $\{\infty\}$ and an $\mathcal F$-open subset of $|\mathcal F|$.


Let $X$ be a topological space such that $|X|=|\mathcal F|$. 
An filter $\mathcal F$ {\em converges} on a topological space $X$ 
iff  one of the two equivalent conditions holds [Bourbaki,II\S7, Def.1]:
\bi\item there is a point $\infty\in X$ such that each $X$-open neighbourhood of $X$ is also $\mathcal F$-open
\item the obvious map $|F| \lra X$ extends to a map $|F|\cup_\FF \{\infty\}\lra X$ %$\mathcal F_\infty^{discrete} \lra X$
\ei

\begin{question}\small\upshape(Ultrafilters and cluster points)
\bee
\item Find a category theoretic way to work with ultrafilters, possibly using that in $Top$, 
for an ultrafilter $\mathcal F$, in $Top$ $\mathcal F \lra \mathcal F\cup_\FF \{\infty\} \rtt g $ 
for each closed map of finite topological spaces, and, more generally, 
$\mathcal F \lra \mathcal F\cup_\FF \{\infty\} \in ({ \{ \{0\}\lra \{0\ra 1\}\}^l}_{finite})^r$ 
(see [Gavrilovich, Lifting property, Claim~1]. 
\item Find a category theoretic way or convenient notation to work with cluster points of filters 
rather than limit point of ultrafilters.
\eee 
\end{question}

A topological space $X$ is {\em quasi-compact} iff  
one of the two equivalent conditions holds [Bourbaki, I\S10.2, Thm.1(d), p.101]:
\bi \item each ultrafilter on $X$ converges
\item for each ultrafilter $\mathcal F$ it holds in $Top$
 $\mathcal F \lra \mathcal F\cup_\FF \{\infty\} \rtt X \lra \{\bullet\}$ 

\item for each ultrafilter $\mathcal F$ it holds in $\sFilt$
 $\ttt(\mathcal F) \lra 	\ttt(\mathcal F\cup_\FF \{\infty\}) \rtt 	\ttt(X) \lra 	\ttt(\{\bullet\})$ 

\ei

More generally, a map $X\lra Y$ is {\em proper} iff for each ultrafilter $\mathcal F$ 
either of the following equivalent conditions holds [Bourbaki, I\S10.2, Thm.1(d), p.101]:
\bi
\item in $Top$, 
 $\mathcal F \lra \mathcal F\cup_\FF \{\infty\} \,\rtt\, X \lra Y$
\item in $\sFilt$,  $\ttt(\mathcal F) \lra 	\ttt(\mathcal F\cup_\FF \{\infty\}) \rtt 	\ttt(X) \lra 	\ttt(Y)$ 
\ei




A {\em Cauchy filter} $\mathcal F$ on a metric space $M$ ([Bourbaki,II\S3.1,Def.2]) 
is a filter on $|M|$ such that one of the two equivalent conditions holds:
\bi\item for each $\epsilon>0$ there is a $\mathcal F$-open non-empty subset $V\subset |M|$ of diameter at most $\varepsilon$ 
\item the map $\mathcal F\times\mathcal F\lra |M|\times |M|$ is continuous where $|M|\times |M|$ is equipped with the topology coming from $\mU(M)$
\item the obvious map $ \Ee(\mathcal F) \lra  \mU (M)$ is well-defined
\ei

A {\em Cauchy sequence} in $M$ is a map $\Ee(\omega_{cofinite})\lra \mU(M)$ where
$\omega_{cofinite}$ is the set of natural numbers equipped with cofinite topology (i.e. a subset is closed iff it is finite).



A metric space is {\em precompact} iff  one of the two equivalent conditions holds :
\bi\item  for each $\varepsilon>0$ there is a finite covering of $M$ by subsets of diameter at most $\varepsilon$ [Bourbaki, II\S4,Thm.3] 
\item each ultrafilter on $M$ is a Cauchy ultrafilter [Bourbaki, II\S4,Exer.5]
\item for each ultrafilter it holds in $\sFilt$ $\iemb (\mathcal F) \lra \Ee(\mathcal F) \rtt \mU (M) \lra \mU(\{\bullet\})$
\ei


A metric space $M$ is {\em complete} iff  one of the two equivalent conditions holds [Bourbaki,II\S3.3,Def.3]:
\bi\item each Cauchy filter on $M$ converges
\item in $\sFilt$, $\Ee (\mathcal F) \lra \Ee(X\cup_\FF \{\infty\}) \rtt \mU (M) \lra \mU(\{\bullet\})$
\item in $\sFilt$, $\ttt(\mathcal F) \lra 	\ttt(X\cup_\FF \{\infty\}) \rtt \mU (M) \lra \mU(\{\bullet\})$
\ei

\begin{question}\small\upshape
Define the completion of a uniform space [Bourbaki, II\S3.7] as something like inner hom 
${\underline{Hom}}(\Ee(\omega_{cofinite}),\mU(M))$. Develop the theory [Bourbaki, II\S3,4] 
of complete and precompact uniform spaces in terms of $\sFilt$ and the lifting properties. 
\end{question}

Let $M$ be a metric space. The following are equivalent [Bourbaki,II\S1.2,Def.3]:
\bi
\item topological space $M_{top}$ is homeomorphic to $|M|$ with the topology induced from the metric on $M$
\item 
there is an arrow  $	\ttt(M_{top})\xra\gamma \mU(M)$ and for each topological space $X$, 
in $\sFilt$ any map $	\ttt(X)\lra \mU (M)$ factors as 
$$	\ttt(X) \lra 	\ttt(M_{top})\xra\gamma \mU (M)$$
\ei

For a compact space $K$, there exists a unique uniform space $K_{uni}$ which induces on $K$ its topology. 
In other words, there is a unique map  $\ttt(K)\xra \gamma \mU(K_{uni})$
such that each map  $\ttt(K)\lra \mU(M)$ factors as $$	\ttt(K)\xra\gamma  \mU(K_{uni}) \lra \mU(M)$$ 
[Bourbaki, II\S4.1,Thm.1].

\begin{remark} The notion of the topology induced by a metric is reminiscent of an adjoint functor to $\ttt$. 
Does either $\mU$ or $\ttt$ have adjoints? 
\end{remark}

\subsubsection{Equicontinuous functions and Arzela-Ascoli theorem}


 Let ${X}$ be a topological space, let ${M}$ be a metric space, and 
let ${(f_i)_{i \in \NN}}$ be a family of functions $f_i:X\lra M$.
%    * We say that this family {f_\alpha} is pointwise bounded if for every {x \in X}, the set {\{ f_\alpha(x): \alpha \in A \}} is bounded in {Y}.
%    * We say that this family {f_\alpha} is pointwise precompact if for every {x \in X}, the set {\{ f_\alpha(x): \alpha \in A \}} is precompact in {Y}.

The family ${f_i}$ is {\em equicontinuous} if either of the following equivalent conditions holds:
\bi
\item for every ${x \in X}$ and ${\epsilon > 0}$, 
there exists a neighbourhood ${U}$ of ${x}$ such that 
${d_Y(f_i(x'), f_i(x)) \leq \epsilon}$ for all ${i \in \NN}$ and ${x' \in U}$
\item the map $\ttt(X)\times \iemb(\{\NN\})\lra \mU(M),\ (x,i)\longmapsto f_i(x)$ is well-defined 
\item the map $\ttt(X)\times \iemb(\NN_{cofinite})\lra \mU(M),\ (x,i)\longmapsto f_i(x)$ is well-defined
\ei

    If ${X = (X,d_X)}$ is also a metric space, we say that the family ${f_i}$ 
is {\em uniformly equicontinuous}  
iff either of the following equivalent conditions holds:
\bi \item for every ${\epsilon > 0}$ there exists a ${\delta > 0}$ such that 
${d_Y(f_i(x'), f_i(x)) \leq \epsilon}$ for all ${i \in \NN}$ and ${x', x \in x}$ with ${d_X(x,x') \leq \delta}$
\item the map $\mU(X)\times \iemb(\{\NN\})\lra \mU(M),\ (x,i)\longmapsto f_i(x)$ is well-defined
\item the map $\mU(X)\times \iemb(\NN_{cofinite})\lra \mU(M),\ (x,i)\longmapsto f_i(x)$ is well-defined
\ei

The family is {\em uniformly Cauchy} iff 
either of the following equivalent conditions holds:
\bi \item for every ${\epsilon > 0}$ there exists a ${\delta > 0}$ and $N>0$ such that 
${d_Y(f_i(x'), f_j(x)) \leq \epsilon}$ for all $i,j>N$ and ${x', x \in x}$ with ${d_X(x,x') \leq \delta}$.
\item the map $\mU(X)\times E(\NN_{cofinite})\lra \mU(M),\ (x,i)\longmapsto f_i(x)$ is well-defined
\ei
Here $\{\NN\}$ denotes the trivial filter on $\NN$ with a unique big subset $\NN$ itself, 
and $\NN_{cofinite}$ denotes the filter of cofinite subsets of $\NN$. 


\begin{question}\small\upshape(Arzela-Ascoli) 
\bee
\item Reformulate various notions of equicontinuity and convergence of a family of functions $f_i:X\lra M$ 
in terms of maps in $\sFilt$
using e.g. $\iemb(\NN_{cofinite})$, $\Ee(\NN_{cofinite})$, $\iemb(\NN_{cofinite}\cup_{\NN_{cofinite}}\{\infty\} )$, 
$\ttt(\NN_{cofinite}\cup_{\NN_{cofinite}}\{\infty\}) $, $E(\NN_{cofinite}\cup_{\NN_{cofinite}}\{\infty\} )$, 
 $\ttt(\NN_{cofinite})$, $\ttt(X)$, $\mU(X)$, and $\mU(M)$. 
\item Reformulate and prove Arzela-Ascoli theorem in terms something like inner $Hom$ in $\sFilt$
and the lifting properties defining precompactness, compactness etc.  
\item Define function spaces in terms of something like inner $Hom$ in $\sFilt$.
\eee
\end{question}



\newpage
\section{Appendix A.}
\section*{Embeddings of geometric categories} 

Here we define several embeddings of geometric categories of metric spaces into the category of ``infinitary'' simplicial objects 
of the category of filters, notably the categroy of metric spaces up to quasi-isometry. 
To make the exposition self-contained, we repeat here some of the notation introduced above.
In part this is motivated by a remark in [Gromov, Hyperbolic dynamics, \S2.7,p.54, footnote 90]. 

It may be interesting to consider here $\FFilt$ instead of $\Filt$. 



Let $\Ord\alpha$ denote the category of finite ordinals less than $\alpha$ and non-decreasing maps;
equivalently but more conceptually, this is the full subcategory of the category of categories 
consisting of the categories $\bullet_0\lra\bullet_1\lra ... $ corresponding to well-ordered sets
of size less than $\alpha$. When $\alpha=\omega+1$, the category $\Ord\omega$ is the category of finite ordinals 
usually denoted $\Delta$. 

For a category $C$ and ordinal $\alpha$, 
{\em $<\!\!\alpha$-simplicial objects in $C$} is a functor $F:\Ord\alpha^{op}\lra C$.
These objects naturally form a category
which we denote  %the category $s_{<\alpha}C=Func(\Ord\alpha^{op},\C)$ 
%of {\em $<\alpha$-simplicial objects in $C$} 
% 
$\salpha C = Func(\Ord\alpha^{op},C)$  of functors from $\Ord\alpha^{op}$ to $C$.
When $\alpha=\omega+1$, this is the usual category of simplicial objects of $C$.

With an object $X$ we can associate two $<\!\!\alpha$-simplicial objects in $C$ as follows.
%$$\iemb_\alpha(X):\beta\mapsto X, \ f \mapsto \id $$ 
$\iemb(X)$ sends each ordinal to $X$ itself and each morphism to the identity
The functor $E_\alpha(X)$ %$E_\alpha(X): \beta\mapsto X^\beta$
sends an ordinal $\beta<\alpha$ to the Cartesian power $X^\beta$, and morphisms are sent to the coordinate maps. 

%However, we find more useful the following modification of $E$:
%for $\beta<\alpha$, $|\Ed(X)(\beta)|=|X|^\beta$ and the topology is defined as follows: 
%a subset is open iff it is either open in the product topology $X^\beta$ 
%or contains the diagonal and a subset of  $X^\beta$ open in the product topology. 
%
 


These two functors define two fully faithful embeddings of $C$ into $s_{<\alpha}C$.
$\iemb:C\lra \salpha C$ and $E:C\lra \salpha C$.

Let $\Filt$ be the category of filters, i.e. the full subcategory of the category of topological spaces consisting of spaces
such that any superset of a non-empty open set is open. 


%Let $\ssTop\alpha=Func(\Ord\alpha^{op},\Topp)$ denote the category of functors from $\Ord\alpha^{op}$
%to $\Topp$. The category $\ssTop\omega$ is the category of simplicial topological spaces. 
%

%In Appendix A we show that this embedding  is implicit in the definition of uniform structures in 
%(Bourbaki, Chapter 2, Uniform Structures). 


\subsection{Metric spaces as ``infinitary'' simplicial filters}

We define several embeddings of categories of metric spaces with various kinds of geometric maps, 
e.g.  uniformly continuous maps,  Lipschitz maps on large scale. We do so by definition 
various filters on (possibly infinite) Cartesian powers of a metric space which preserve 
certain geometric information about the metric space. In the usual way these collections of filters
give rise to simplicial objects of $s_{\leq \omega}\Filt$.  

 

Let $M$ be a metric space. Let us now define a number of topologies on 
Cartesian powers of $|M|$. 

\bi
\item A non-empty subset of $M^n$ is $\tau$-open (big) iff the following formula holds:
 $\forall x_1\in M \exists U_{x_1}\ni x_1  \text{ where }U_{x_1} \text{ is a neighbourhood of } x_1 \\ 
\forall x_2\in U_{x_1} \exists U_{x_2} \ni x_2 \text{ where }U_{x_2} \text{ is a neighbourhood of } x_2 \\
....\\
  \forall x_n\in U_{x_{n-1}} \exists  U_{x_n} \ni x_n \text{ where  }U_{x_n} \text{ is a neighbourhood of } x_n \\ 
 \text{ \ } (x_1,x_2,...,x_n) \in U  $

%
%Motivation. In Bourbaki, Introduction they say topology enables us to say 
%a property holds for all points y 'as near x as we please', 'sufficiently near x'. 
%In topological arguments we iterate it, i.e.
%consider y as near x as we please, 
%consider z as near y as we please, 
%etc... and the topology/simplicial structure on XxXx...xX is to capture this. 
%

\item 
A non-empty subset of $M^n$ 
is $\tau_U$-open iff it contains an $\varepsilon$-neighbourhood of the diagonal $\{\, (x,x,...,x)\,:\, x\in |M|\,\}$
for some $\varepsilon>0$,~i.e.
$U\subseteq |M|^n$ is open iff there is $\varepsilon>0$ such that for each $x_1,...,x_n\in M$,
it holds
$(x_1,...,x_n)\in U$
provided there is  $x\in M$ such that %, each $x_1,...,x_n\in M$,
$dist(x,x_i)<\varepsilon$, $i=1,..,n$. % implies $(x_1,...,x_n)\in U$. 


Note that no proper subset $\emptyset\subsetneq U\subsetneq |M|$ is open %is antidiscrete, 
as you may take $x=x_1$.



\item  Fix a real number $D>0$. 
A non-empty subset of $M^n$ 
is $\tau_D$-open iff it contains an $D$-neighbourhood of the diagonal $\{\, (x,x,...,x)\,:\, x\in |M|\,\}$
for some $\varepsilon>0$,~i.e.
$U\subseteq |M|^n$ is open iff there  for each $x_1,...,x_n\in M$,
it holds
$(x_1,...,x_n)\in U$
provided there is  $x\in M$ such that 
$dist(x,x_i)\leq D$, $i=1,..,n$. 

Note that no proper subset $\emptyset\subsetneq U\subsetneq |M|$ is open %is antidiscrete, 
as you may take $x=x_1$.




\item A non-empty subset of $M^\omega$ 
is $\tau_{\mathcal L}$-open 
iff there is $\lambda>0$, $N>0$, $D< \lambda N$  such that for each $x_1,...,x_n,...\in M$,
it holds
$(x_1,...,x_n,...)\in U$
provided there is  $x\in M$ such that
$x=x_1=...=x_N$ and
$dist(x,x_i)\leq \lambda i - D $ for each $i>N$.

A non-empty subset $U$ of $M^n$ is $\tau_{\mathcal L}$-open iff 
it contains the diagonal $\{ (x,..,x) : x \in M \}$.


\item A non-empty subset of $M^\omega$ 
is $\tau_{\mathcal L_1}$-open 
iff there is  $N>0$, $ D<N$ such that for each $x_1,...,x_n,...\in M$,
it holds
$(x_1,...,x_n,...)\in U$
provided there is  $x\in M$ such that
$x=x_1=...=x_N$ and
$dist(x,x_i)\leq i-D $ for each $i>N$.
%$dist(x,x_i)\leq min(0,i-N) $ for each $i>N$.


A non-empty subset $U$ of $M^n$ is $\tau_{1}$-open iff 
it contains the diagonal  $\{ (x,..,x) : x \in M \}$.

\ei



A map $f:|M|\lra |N|$ induces a map $f_n:|M|^n\lra |N|^n$. 
The following is easy to check:
\bi

\item For $n>1$, 
$f_n$ is 
is $\tau$-continuous iff it is continuous.


\item For $n>1$, 
$f_n$ is 
is $\tau_U$-continuous iff it is uniformly continuous.


\item For $n>1$,
$f_n$ is $\tau_D$-continuous iff for each $x,y\in M$ 
$dist(x,y) \leq D$ implies $dist(f(x),f(y)) \leq D$




\item 
$f_\omega$ is $\tau_{\mathcal L}$-continuous iff it is $\lambda$-Lipschitz on large scale  
for some $\lambda,D>0$,~i.e.
$dist(f(x),f(y))\leq \lambda dist(x,y)$ whenever $dist(x,y)\geq D$, $x,y\in M$.


\item 
$f_\omega$ is $\tau_{\mathcal L_1}$-continuous iff it is $1$-Lipschitz on 
large scale,~i.e.for some $D$ for each $x,y\in M$
$dist(f(x),f(y))\leq dist(x,y)+D$ 

\ei

A map $f:M\lra M$ is {\em an almost isometry} iff 
either of the following equivalent conditions holds:
\bi
\item $dist(f(x),f(y))\leq dist(x,y)+D$ for some $D$
%\item $f_2:M\times M\lra M\times M$ is $\tau_D$-continous for some $D$. 
\item $f_\omega:M^\omega\lra M^\omega$ is $\tau_{\mathcal L_1}$-continuous
\ei



A map $f:M\lra M$ is {\em a quasi-isometry} iff 
either of the following equivalent conditions holds:
\bi
\item $dist(f(x),f(y))\leq \lambda dist(x,y)+D$ for some $\lambda,D>0$ for each $x,y\in M$
%\item $f_2:M\times M\lra M\times M$ is $\tau_D$-continous for some $D$. 
\item $f_\omega:M^\omega\lra M^\omega$ is $\tau_{\mathcal L}$-continuous
\ei


A verification shows that these topologies define fully faithful functors
$$m_{\mathcal U}: \MUniform \lra Func(\Ordfiniop, \Filt),$$
$$m_{ D}: \MDadd \lra Func(\Ordfiniop, \Filt),$$
$$m_{\mathcal L}: \MLipschitz\lra Func(\Ord{\omega+1}^{op}, \Filt),$$
$$m_{\mathcal L_1}: \MLipschitzOne \lra Func(\Ord{\omega+1}^{op}, \Filt) $$
from the relevant geometric categories of metric spaces. 

%%%
%%%\subsection{Metric spaces with uniformly continious maps as simplicial topological spaces}
%%%
%%%We use terminology of (Gromov, Hyperbolic dynamics). 
%%%
%%%We define a fully faithful embedding $m_l:\qLips\lra\ssTop{\omega+1}$ of the category of metric spaces with uniformly continious maps
%%%into the category $\ssTop{\omega+1}$.
%%%
%%%Let $M$ be a metric space. For each $n\in\omega$ finite let $M_n$ be denote the set of points $M^n$
%%% equipped with antidiscrete topology. 
%%%
%%%
%%%Let $M_\omega$ be the set of points $|M|^\omega$ equipped with the following topology.
%%%A subset $U\subseteq |M|^\omega$ of $M^\omega$ 
%%%is {\em open}
%%%iff there is $\lambda>0$, $N>0$ such that for each $x_1,...,x_n,...\in M$,
%%%it holds
%%%$(x_1,...,x_n,...)\in U$
%%%provided there is  $x\in M$ such that %, each $x_1,...,x_n\in M$,
%%%$dist(x,x_i)<\lambda i$ for each $n>N$.
%%%
%%%In an obvious way this gives a functor $M_\bullet:\Ord{\omega+1} \lra\Topp$ with help of 
%%%the diagonal embeddings and projection maps. 
%%%To see this, note that continuity poses no restriction on maps $M_\omega\lra M_n$ as the topology on $M_n$ is antidiscrete, 
%%%and any monotone map $M_n\lra M_\omega$ is eventually constant.
%%%A map $f:|M|\lra |N|$ induces a possibly discontinous map $f_\omega:M_\omega\lra N_\omega$. This map is  continious 
%%%iff $f$ is a large scale Lipschitz map, i.e. there is $\lambda>0$,$D>0$ such that
%%% $dist(f(x),f(y)) \leq  \lambda dist(x,y)$ whenever $dist(x,y)\geq D$. 
%%%
%%%
%%%
%%%
%%%
%%%\subsection{Metric spaces with uniformly continious maps as simplicial topological spaces}
%%%
%%%We use terminology of (Bourbaki, Chapter 2, Uniform Structures). 
%%%
%%%We reformulate several notions in terms of the embedding 
%%%$m_{\mathcal U}: \MUniform \lra Func(\Ordfiniop, \Topp)$.
%%%
%%%\subsubsection{Compact and complete metric spaces}
%%%
%%%A {\em filter} is a topological space $\mathcal F$ such that (i) a superset of a non-empty open set is open 
%%%(ii) the intersection of two non-empty open sets is open. 
%%%An {\em ultrafilter} is a filter such that if the union of finitely many open sets is open, then one of them is.
%%%
%%%With a filter $\mathcal F$ on the set of points of a topoolical space $X$ 
%%%associate a topological space $X\cup_\FF \{\infty\}$ such that $\FF$ is the neighbourhood filter of $\infty$:
%%%$|X\cup\FF \{\infty\}|=|X|\cup\{\infty\}$, and a subset is open iff it is either an open subset of $X$
%%%or contains $\infty$ and is a union of $\{\infty\}$ and an open subset of $X$ which is also $\FF$-open.
%%%
%%%%  associate two topologies  $\mathcal{F_1}$ and $\mathcal{F}_\infty^{discrete}}$
%%%%  on the set $|\mathcal{F_1}|=|\mathcal F| \cup \{\infty\}$ as follows.
%%%%  A subset of $|\mathcal F| \cup \{\infty\}$ is $\mathcal{F_1}$-open if it is either a $\mathcal F$-open subset of $|\mathcal F|$ or 
%%%%  a union of $\{\infty\}$ and an $\mathcal F$-open subset of $|\mathcal F|$.
%%%%  
%%%%  A subset of $|\mathcal F| \cup \{\infty\}$ is $\mathcal{F_1}^{discrete}$-open if it is either a subset of $|\mathcal F|$ or 
%%%%  a union of $\{\infty\}$ and an $\mathcal F$-open subset of $|\mathcal F|$.
%%%
%%%
%%%Let $X$ be a topological space such that $|X|=|\mathcal F|$. 
%%%An filter $\mathcal F$ {\em converges} on a topological space $X$ 
%%%iff  one of the two equivalent conditions holds:
%%%\bi\item there is a point $\infty\in X$ such that each $X$-open neighbouhood of $X$ is also $\mathcal F$-open
%%%\item the obvious map $|F| \lra X$ extends to a map $|F|\cup_\FF \{\infty\}\lra X$ %$\mathcal F_\infty^{discrete} \lra X$
%%%\ei
%%%
%%%
%%%A topological space $X$ is {\em compact} iff  
%%%one of the two equivalent conditions holds [Bourbaki, General Topology, I\S10.2, Thm.1(d), p.101]:
%%%\bi \item each ultrafilter on $X$ converges
%%%\item for each ultrafilter $\mathcal F$ it holds
%%% $\mathcal F \lra \mathcal F\cup_\FF \{\infty\} \rtt X \lra \{\bullet\}$ 
%%%\ei
%%%
%%%
%%%
%%%
%%%A {\em Cauchy filter} $\mathcal F$ on a metric space $M$ ([Bourbaki,II\S3.1,Def.2]) 
%%%is a filter on $|M|$ such that one of the two equivalent conditions holds:
%%%\bi\item for each $\epsilon>0$ there is a $\mathcal F$-open non-empty subset $V\subset |M|$ of diameter at most $\varepsilon$. 
%%%\item the map $ \Ed(\mathcal F) \lra  \mU (M)$ is well-defined
%%%\ei
%%%
%%%A {\em Cauchy sequence} in $M$ is a map $\Ed(\NN_{cofinite})\lra \mU(M)$ where
%%%$\NN_{cofinite}$ is the set of natural numbers equipped with cofinite topology (i.e. a subset is closed iff it is finite).
%%%
%%%
%%%
%%%A metric space is {\em precompact} iff  one of the two equivalent conditions holds:
%%%\bi\item  for each $\varepsilon>0$ there is a finite covering of $M$ by subsets of diameter at most $\varepsilon$. 
%%%\item each ultrafilter on $M$ is a Cauchy ultrafilter [Bourbaki, I\S4,Ex.5]
%%%\item for each ultrafilter it holds $\iemb (\mathcal F) \lra \Ed(\mathcal F) \rtt \mU (M) \lra \iemb(\{\bullet\})$
%%%\ei
%%%
%%%
%%%A metric space is {\em complete} iff  one of the two equivalent conditions holds:
%%%\bi\item each Cauchy filter converges
%%%\item $\Ed (\mathcal F) \lra \Ed(\mathcal F_\infty) \rtt \mU (M) \lra \iemb(\{\bullet\})$
%%%\ei
%%%
%%%Let $M$ be a metric space. The following are equivalent:
%%%\bi
%%%\item $M_{top}$ is homeomorphic to $|M|$ with the topology induced from the metric on $M$
%%%\item 
%%%there is an arrow  $\Ed(M_{top})\xra\gamma \mU(M)$ and for each topological space $M'$, any map $\Ed(M')\lra \mU (M)$ factors as 
%%%$$\Ed(M') \lra \Ed(M_{top})\xra\gamma \mU (M)$$
%%%\ei
%%%
%%%For a compact space $K$, there exists a uniform space $K_{uni}$ and a map $\Ed(K)\xra \gamma \mU(K_{uni})$
%%%such that each map  $\Ed(K)\lra \mu(M)$ factors as $$\Ed(K)\xra\gamma  \mU(K_{uni}) \lra \mU(M)$$.
%%%
%%%
%%%
%%%\subsubsection{Completetion and induced topology}
%%%
%%%
%%%
%%%For two topological spaces $X,Y$, define {\em open-open filter} topology on $Hom(X,Y)$:
%%%a subset is open iff it contains
%%%$O_{U,V}:=\{ f:X\lra Y \,:\, f(U) \subset V\ \}$
%%%for some  $U\subset X, V\subset Y$ open subsets of $X$ and $Y$.
%%%Denote by $\hhom(X,Y)$ the set  $Hom(X,Y)$ equipped with this topology.
%%%
%%%For two objects $X,Y\in \Ob \ssaTop=Func(\Ord\alpha,\Topp)$, 
%%%$Hom(X,Y)$ is the set of natural transformations $\gamma:X\implies Y$.
%%%Open-open topology on Hom-sets lets us define a simplicial object corresponding 
%%%to $Hom(X,Y)$: $\hhom(X,Y)(\beta)$ is the subset 
%%%$\{ \gamma_\beta:X(\beta)\lra Y(\beta) \ :\ \gamma\in Func(X,Y)\ \}\subset Hom(X(\beta),Y(\beta))$
%%%equipped with the topology induced from the open-open topoloy on $Hom(X(\beta),Y(\beta)$.
%%%The morhpisms are the coordinate maps. Denote by $\hhom(X,Y)$ the $<\!\!\alpha$-simplicial object so defined.
%%%
%%%The completion of a metric space $M$ is
%%%$\hhom(\Ed(\NN_{cofinite}), \mU(M))$.
%%%
%%%A sequence of equicontinous functors from a topological space $X$ to a metric space $M$
%%%is a map 
%%%$$\Ed(X)\times \Ed(\NN_{cofinite}\lra \mU(M)$$.
%%%The object
%%%$$\hhom ( \Ed(X)\times \Ed(\NN_{cofinite},  \mU(M)) $$
%%%may be thought of as a space of such sequences (FIXME: add detail, Arzela-Ascoli theorem etc). 
%%%
%%%========
%%%




\newpage
\section{Appendix B.} 

\section*{Reading Bourbaki definition of the uniform spaces}

(Bourbaki, II\S1.1.1) treats metric spaces as uniform spaces; we observe that 
the uniform space is a simplicial object. 
%\footnote{ FIXME: write Kolya's observation
%that this simplicial object is ``a set factored by $dist(x,y)<\epsilon"$
%(which isn't an equivalence relation) in some higher homotopy sense. 
%Alas, i dont quite understand this argument...}


We quote (Bourbaki, I\S6.1.1) and  (Bourbaki, II\S1.1.1):
\begin{quote}
DEFINITION I. A filter on a set $X$ is a set $\mathcal F$ of subsets of $X$ which has the
following properties:\\
$(F_I)$ Every subset of $X$ which contains a set of F 
 belongs to 
$\mathcal F$.\\
$(F_{II})$ Every finite intersection of sets of 
$\mathcal F$ belongs to 
$\mathcal F$.\\
$(F_{III})$ The empty set is not in 
$\mathcal F$.
\end{quote}

\begin{quote}
%Uniform Structures I. UNIFORM SPACES
1. DEFINITION OF A UNIFORM STRUCTURE
\vskip 1pt\noindent 
DEFINITION I. {\em A uniform structure (or uniformity) on a set $X$ is a structure
given by a set $\mathfrak U$ of subsets of $X\times X$ which satisfies axioms $(F_I)$ and $(F_{II})$
of Chapter I,
 6, no. I and also satisfies the following axioms:\\
$(U_I)$ Every set belonging to $\mathfrak U$ contains the diagonal $\Delta$.\\
 $(U_{II})$ If $V \in \mathfrak U$ then $V^{-1} \in \mathfrak U$.\\
 $(U_{III})$ For each $V \in\mathfrak U$ there exists $W \in \mathfrak U$ such that $W\circ W \subset V$.\\
The sets of $\mathcal U$ are called entourages of the uniformity defined on $X$ by $\mathcal U$.
A set endowed with a uniformity is called {\em a uniform space}.
If $V$ is an entourage of a uniformity on $X$, we may express the
relation $(x, x') \in V$ by saying that ``$x$ and $x'$ are $V$-close".
}\end{quote}

The set of points of a metric space $X$ carries a canonical uniform space: $V\in\mathfrak U$ iff $\{(x,x')\,:\,dist(x,x')<\varepsilon\}\subset V$ for some $\varepsilon>0$. 

\def\fTop{\sFilt}
Let us translate the definitions above to the language of arrows: we shall see that
a uniform space may be viewed as a simplicial object of the category 
of topological spaces.

First notice that a filter can equivalently be defined as a non-discrete topology such that a superset of a non-empty open set is necessarily open:
a filter $\mathcal F$ on a set $X$ defines a topology on $X$
where a subset is open iff it is either $\mathcal F$-big or empty.  Indeed, 
Axioms $(F_I)$ and $(F_{II})$ of a filter imply that the family of subsets $\mathfrak U\cup\{\emptyset\}$ 
is a topology on a set  $X$. 


In this way a uniform structure on a set $X$ defines a %filter 
topology on $X\times X$.


Axiom $(U_I)$ implies that the diagonal map $X\xra{(x,x)} X\times X$ is continuous as 
a map from the set $X$ equipped with antidiscrete topology to the set $X\times X$ equipped
with the topology above, and is almost equivalent to this. Indeed, the latter says that
an $\mathfrak U$-big subset of $X\times X$ either contains the diagonal or does not intersect it. 



Axiom $(U_{II})$ says that 
permuting the coordinates $X\times X\lra X\times X, (x_1,x_2)\mapsto (x_2,x_1)$
is continuous in this topology.  


Define topology on the set $X\times X\times X$ via the pullback square in the category of filter topological spaces $\fTop$:
 $$ \xymatrix{ {X\times X\times X} \ar[r]^{(p_1\times p_2)} \ar[d]_{(p_2\times p_3)} & {X\times X} \ar[d]^{p_2} \\
{X\times X } \ar[r]_{p_2}  & { X } }$$



Axiom $(U_{III})$ says that the map  $X\times X\times X \xra{(p_1,p_3)} X\times X, (x_1,x_2,x_3)\mapsto (x_1,x_3)$
is continuous in this topology. Indeed, by definition
$$W_1\circ W_2= \{(x_1,x_3) \,:\, (x_1,x_2)\in W_1,\,\, (x_2,x_3)\in W_2\}$$ 
and the sets of form
$$\{(x_1,x_2,x_3) \,:\, (x_1,x_2)\in W_1,\,\, (x_2,x_3)\in W_2\}
=W_1\times X\cap X\times W_2,$$ 
$W_1,W_2\in \mathfrak U$,
form a base of the pullback topology on $X\times X\times X$. Hence, $(U_{III})$ says 
that the preimage of an open subset of $X\times X$ under $(p_1,p_3)$ contains an 
open subset of $X\times X\times X$, i.e.~is open (as pullback is taken among filter topologies).  


Axiom $(U_I)$ implies that the diagonal map $X\xra{(x,x)} X\times X$ is continuous as 
a map from the set $X$ equipped with antidiscrete topology to the set $X\times X$ equipped
with the topology above.  

 Note that $W\circ W$ intersects the diagonal and the continuity of the diagonal map $X\xra{(x,x)} X\times X$
 implies $W\circ W$ contains the diagonal. Thus, in presence of $(U_{III})$, 
$(U_{I})$ is equivalent to the continuity of the diagonal map  $X\xra{(x,x)} X\times X$ 
in the topologies indicated.


Let $X_1$ denote the set $X$ equipped with the antidiscrete topology.
Let $X_2$ and $X_3$ denote the sets $X\times X$ and $X\times X\times X$ equipped with the topologies above.
For $n>3$, let $X_n$ be the pullback in $\fTop$ 
 $$ \xymatrix{ {X_n} \ar[r]^{(p_2\times\ldots\times p_n)} \ar[d]_{(p_1\times p_2)} & {X_{n-1}} \ar[d]^{p_2} \\
{X_2 } \ar[r]_{p_2}  & { X } }$$

The axioms above ensure that the ``set-theoretic'' face and degeneracy maps  
$$(p_{i_1},...,p_{i_k}):X\times ... \times X  \lra X\times ... \times X$$ 
are continuous. Thus we see that a uniform structure on a set $X$ defines
a simplicial complex $X_n$ in $\fTop$, 
$$(p_{i_1},...,p_{i_k}):X_n \lra X_m$$ 

\begin{claim} A uniform structure on a set $X$ is a simplicial object $X_\cdot$ in the subcategory $\fTop$ of 
filter topological spaces
equipped with an involution $i:X_\cdot\lra X_\cdot$
such that
\bi
\item $X_1$ is the set $X$ equipped with antidiscrete topology
%\item $X_2$ is the set $X\times X$ equipped with filter topology
\item the underlying set of $X_2$ is $X\times X$
\item $i:X_{\cdot} \lra X_\cdot$ is the involution permuting the coordinates on $X\times X$
\item for $n>2$, $X_n$ is the pullback as described above 
\ei
\end{claim}

\begin{question}\small\upshape %FIXME: is it obviously true that 
%this simplicial object has some universality property,
%e.g. the pullback property is equivalent to ``there is a unique map from any other simplicial object
%with the same $X_1$ and $X_2$", maybe with ``some $S_n$ group action"  ? is it called being some kinda 2-coskeleton ?  
Find a categorical description of the simplicial objects obtained from uniform spaces.
\end{question}






Acknowledgements. To be written. 


 This work is a continuation of [DMG]; early history is given there. I thank
M.Bays, D.Krachun, K.Pimenov, V.Sosnilo, S.Synchuk and P.Zusmanovich for discussions and proofreading; I
thank L.Beklemishev, N.Durov, S.V.Ivanov, S.Podkorytov, A.L.Smirnov for discussions. I also
thank several students for encouraging and helpful discussions.  Chebyshev
laboratory, St.Petersburg State University, provided a coffee machine and an
excellent company around it to chat about mathematics.  Special thanks are to
Martin Bays for many corrections and helpful discussions. Several observations
in this paper are due to Martin Bays. I thank S.V.Ivanov for several
encouraging and useful discussions; in particular, he suggested to look at the
Lebesque's number lemma and the Arzela-Ascoli theorem. A discussion with Sergei
Kryzhevich motivated the group theory examples.

   Much of this paper was done in St.Petersburg; it wouldn't have been possible
without support of family and friends who created an excellent social
environment and who occasionally accepted an invitation for a walk or a coffee
or extended an invitation; alas, I made such a poor use of it all.  


  This note is elementary, and it was embarrassing and boring, and
embarrassingly boring, to think or talk about matters so trivial, but luckily
I had no obligations for a time.


\newpage



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\end{thebibliography}



\end{document}

