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%%%%%%%%%%%%%%%%
%%%
%%%\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.
%}}
%}
%%\setcounter{secnumdepth}{4}
%\renewcommand{\thesubsubsection}{\arabic{subsubsection}}
%%\newcounter{para}
%%\newcommand\mypara[1]{\par\refstepcounter{para}{{\l\thepara}\space\textbf{#1\space}}}
\begin{document}
\selectlanguage{english}\catcode`\_=8\catcode`\^=7 \catcode`\_=8
\def\mmu{{\ethi{\ethmath{'se}}}}
\def\mmufini{{{\ethi{\ethmath{'se}}}_{\!\!\!<\omega}}}
%
\date{}%9 May 2017}
\title[A naive approach to tame topology]{
%%%%n
A naive diagram-chasing approach to formalisation of tame topology
%%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 and konstantin pimenov\thanks
{\tiny A draft of a research proposal.
Comments welcome at
$\tt{mi\!\!\!ishap\!\!\!p@sd\!\!\!df.org}$.
I thank Martin Bays, Sergei Ivanov and Vladimir Sosnilo for discussions. {\tiny\href{http://mishap.sdf.org/mintsGE/}{http://mishap.sdf.org/mints/}}
updated: 15.9.18, 18.10.18% 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.
}%\\
%\small { a draft of a research proposal}
\\
{\sf\small in memoriam: evgenii shurygin}\\
%%\\
%%\small { a draft of a research proposal}
%%%--- \\
%%%--- {\sf\small in memoriam: evgenii shurygin}\\
%%%--- \\\tiny%\fontfamily{c}
%%%--- %In the following sections we bring up
{\small---------------------------------------------------------------------------------------------------------} \\
%%%--- % instances %%of human and animal behavior
%%%--- % which are, on one hand,
%%%--- %% [...] miraculously complicated,
%%%--- %on the other hand
%%%--- %%[...] they have
%%%--- %% 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
%%%--- %\rmseries\fontfamily{c}
%%%--- %\selectfont %instances
\tiny
{\tiny ...~due to internal constraints on
possible architectures of unknown to us~functional~"mental~structures".
}\\ {\tiny
Misha Gromov. Structures, Learning and Ergosystems: Chapters 1-4, 6.
} %\\
%%%--- %\small { a draft of a research proposal}
%%%--- %\\
%%%--- %{\sf\small 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}
We rewrite excerpts of [Bourbaki, General Topology] %classical topological definitions
using the category-theoretic %? diagram chasing ?
notation of arrows
and are thereby led to %their
concise reformulations
of classical topological definitions
in terms of
simplicial categories and
orthogonality of morphisms, which we hope might be of use
in the formalisation of topology and in developing the tame topology of Grothendieck.
Namely, we observe that topological and uniform spaces are simplicial objects
in the same category, a %?the?
category of filters or, equivalently, the category of pointed topological spaces with maps continuous at the point, and
that a number of elementary properties can be
obtained by repeatedly passing
to the left or right orthogonal (in the sense of Quillen model categories) %$C^l, C^r, C^{lr}, C^{ll}, C^{rl}, C^{rr},...$
starting from a simple class of morphisms, often
a single typical (counter)example appearing implicitly in the definition.
Examples include the notions of: compact, discrete, connected, and
totally disconnected spaces, dense image, induced topology, and separation axioms,
and, outside of topology, finite groups being
nilpotent, solvable, torsion-free, $p$-groups, and prime-to-$p$ groups;
injective and projective
modules; injective and surjective (homo)morphisms.
%%%---
%%%---
%%%--- ...they are instances of several standard constructions, orthogonality of morphisms
%%%---
%%%--- A number of elementary properties can be obtained by repeatedly passing
%%%--- to the left or right orthogonal $C^l, C^r, C^{lr}, C^{ll}, C^{rl}, C^{rr},...$
%%%--- starting from a simple class of morphisms, often
%%%--- a single (counter)example to the property you define.
%%%---
%%%--- The counterexample is often implicit in the text of the definition of the property.
%%%---
%%%---
%%%--- and thereby reformulate them in terms of category theory.
%%%--- observe this leads to a reformulation in terms of seve
%%%---
%%%---
%%%---
%%%--- and observe this leads to a reformulation?interpretation of these definitions in terms of category theory.
%%%---
%%%---
%%%--- ...We show how an unsophisticated, almost mechanical attempt to rewrite excerpts of
%%%--- text of \href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki, General topology]} using notation of arrows
%%%--- and notions of category theory leads to
%%%--- reformulations of certain elementary notions of topology
%%%--- in terms of category theory, and that these reformulations correspond
%%%--- more directly to the usual intuition and text of elementary definitions.
%%%--- ....
%%%--- %\text{{\ethi evgenii shurygin nI gA gI gE ga gi ge .cE ge ne }}
%%%---
%%%---
%%%---
%%%---
%%%---
%%%---
%%%---
%%%---
%%%---
%%%---
%%%---
%%%---
%%%---
%%%---
%%%--- %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}\vskip-11pt
\setcounter{tocdepth}{4}
\vskip-11pt
\tableofcontents
\begin{flushright}
%\begin{center}
\tiny
\begin{minipage}[t]{0.7\textwidth}
\tiny {{\tiny Die Mathematiker sind eine Art Franzosen: Redet man zu ihnen, so
\"ubersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas
anderes.\ }
\tiny\ \ \,\,\,\,\, ------ Johann Wolfgang von Goethe. Maximen und Reflexionen. Nr.~1005.
%Aphorismen und Aufzeichnungen. %Nach den Handschriften des Goethe- und Schiller-Archivs hg. von Max Hecker, Verlag der Goethe-Gesellschaft,
%Weimar 1907,
%Aus dem Nachlass, Nr. 1005, Uber Natur und Naturwissenschaft.
}\end{minipage}
\footnote
{
Mathematicians are like Frenchmen: whatever you say to them they translate
into their own language, and forthwith it is something entirely different.
\ In:
Johann Wolfgang von Goethe. Aphorismen und Aufzeichnungen. Nach den Handschriften des Goethe- und Schiller-Archivs hg.~von Max Hecker, Verlag der Goethe-Gesellschaft,
Weimar 1907.
Aus dem Nachlass, Nr.~1005, Uber Natur und Naturwissenschaft. Maximen und Reflexionen.
}
\vskip5pt
\tiny
\begin{minipage}[t]{0.73\textwidth}
\begin{minipage}[t]{0.958\textwidth}
%Arguments from analogy are always at hand, and grow
%up spontaneously in a fruitful imagination, while argu-
%ments that are more direct, and more conclusive, often
%require painful attention and application : and therefore,
%mankind in general have been very much disposed to trust
%to the former. If one attentively examines the systems
%of the ancient philosophers, either concerning the mate-
%rial world, or concerning the mind, he will find them to
%be built solely upon the foundation of analogy. Lord Ba-
%con first delineated the strict and severe method of induc-
%tion ; since his time it has been applied with very happy
%
%vol. I. 54
%
%
%
%426 or THE HUMAN MINU.
%
%success in some parts of natural philosophy ; and bardir
%in any thing else. But there is no subject in which man-
%kind are so much disposed to trust to the analogical way
%of thinking and reasoning, as in what concerns the mind
%and its operations ; because, to form clear and distinct
%notions of those operations in the direct and proper way,
%and to reason about them, requires a habit of attentive
%reflection, of which few are capable, and which, even by
%those few, cannot be attained without much pains and
%labour.
%
%Every man'is apt to form his notions of things difficult
%to be apprehended, or less familiar, from their analogy
%to things which are more familiar. Thus,
if a man bred
to the seafaring life %,
...
%and accustomed to think and talk
%only of matters relating to navigation,
%enters into dis-
%course upon any other subject ; it is well known, that the
%language and the notions proper to his own profession are
%infused into every subject, and all things are measured by
%the rules of navigation :
%and if he
should take it into his
head to philosophize concerning the faculties of the mind,
it cannot be doubted, but he would draw his notions from
the fabric of his ship, and would find in the mind, sail,
masts, rudder, and compass.
%
%Sensible objects of one kind or other, do no less occupy
%and engross the rest of mankind, than things relating to
%navigation, the seafaring man. For a considerable part
%of life, we can think of nothing but the objects of sense j
%and to attend to objects of another nature, so as to form
%clear and distinct notions of them, is no easy matter, even
%after we come to years of reflection. The condition of
%mankind, therefore, affords good reason to apprehend,
%that their language, and their common notions, concern-
%ing the mind and its operations, will be analogical, and
%derived from the objects of sense 5 and that these analo-
%gies will be apt to impose upon philosophers, as well as
%upon the vulgar, and to lead them to materialize the mind
%and its faculties,- and experience abundantly confirms the
%truth of this.
%
\\ \tiny%\hskip12pt \ \ \ \ \ \ \ \ \,\,\,\,\, ----------------------
------ Thomas Reid. An Inquiry into the Human Mind on the Principles of Common Sense. 1764.
\end{minipage}
\end{minipage}
\end{flushright}
\section{
%%%%
Introduction.
}
\subsection{Main ideas.}
In this note we rewrite several classical definitions and constructions in topology in terms
of category theory and diagram chasing. We do so by
by first ``transcribing'' excerpts of \href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki, General Topology]} and [Engelking, Topology]
by means of notation extensively using arrows, and then recognizing familiar patterns of
standard category-theoretic constructions and diagram chasing arguments.\footnote{
Arrows and other category-theoretic notations are conspicuously absent from [Bourbaki, General Topology]
and little used in his other books. \href{http://www.tau.ac.il/~corry/publications/articles/pdf/bourbaki-structures-synthese.pdf\#338}{[Corry, Nicolas Bourbaki and the Concept of Mathematical Structure]}, also [Dieudonn\'e, The work of Bourbaki during the last thirty years] might suggest that Bourbaki conciously avoided category-theory notation.
}
Arguably,
{\em we transcribe the ideas of Bourbaki into a language of category theory
appropriate to these ideas}, and our analysis of the text of Bourbaki
shows these ideas (but not notation) are implicit in Bourbaki and reflect their
logic (or perhaps their ergologic in the sense of %[Gromov, Ergobrain]
\href{http://www.ihes.fr/~gromov/PDF/ergobrain.pdf}{[Gromov. Ergobrain; }
\href{http://www.ihes.fr/~gromov/PDF/ergo-cut-copyOct29.pdf}{Memorandum Ergo]}).
Doing so, we observe that a number of elementary textbook properties
are obtained by taking the orthogonal (in the sense of Quillen lifting property)
to the simplest morphism-counterexample, and this
leads to a concise syntax expressing these properties in two or three bytes
in which e.g. denseness, separation property Kolmogoroff/$T_0$, compactness is expressed as
%$$ (\{ c \}\lra \{ o \ra c \})^l \ \ \ \ (\{x\leftrightarrow y \}\lra \{ x =y \})^r\ \ \ \ \ ((\{ \{o\}\lra \{o\ra c\}\}^{r})_{<5})^{lr} $$
$$
\begin{array}{ccccccc}
\text{(dense image)}
& &
\text{(Kolmogoroff/$T_0$)}
& &
\text{(compact)}
\\
(\{ c \}\lra \{ o \ra c \})^l &\ \ \ \ &
(\{x\leftrightarrow y \}\lra \{ x =y \})^r & \ \ \ \ \ &
((\{ \{o\}\lra \{o\ra c\}\}^{r})_{<5})^{lr}
\end{array}
$$
this shows their \href{https://arxiv.org/abs/1301.0081}{Kolmogoroff complexity} is very low (byte or two).
We also observe that the categories of topological spaces, 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 spaces of large scale metric geometry are also simplicial objects of a category of filters with different morphisms.
This is, moreover, implicit in the definitions of a topological, uniform, and coarse space.
The exposition is in form of a story where we pretend to ``read off''
category-theoretic constructions from the text of excerpts of [Bourbaki] and
[Engelking] in a straightforward, unsophisticated, almost mechanical manner.
We hope word ``mechanical'' can be taken literally: we pretend to search for
correlations between the structure of allowed category-theoretic
diagram-chasing constructions and the text of arguments in topology, and hope
this search can be done by a short program.
No attempt is made to develop a theory or prove a theorem: our goal is
to explain the process of transcribing by working out a few
examples in detail. In fact,
we think that understanding and formalising this process %of transcribing (``reading off')'
is a very interesting question.
This note is a research proposal suitable for a polymaths project:
%little attempt is made to prove theorems,
% we try to formulate rather than answer any questions which arise.
%We feel this proposal suits a polymaths project. Indeed,
transcribing topological arguments into category theory involves rather independent tasks:
finding topological arguments worth transcribing and working out the precise meaning of category theoretic reformulations
are best suited for general topologists; spotting category theoretic patterns is best suited for category theorists;
working out formal syntax is best suited for logicians.
We hope our way of translating might of use in the formalisation of topology
and suggests an approach to
the tame topology of Grothendieck.
\subsection{Contents.} In \S\ref{intro:sur}, as a warm-up and an example of our translation, we discuss the definition of surjection;
in \S\ref{yoga:ort}, we suggest the intuition that orthogonality is category-theoretic {\em negation}.
Appendix~\S\ref{surinj} gives a verbose exposition of the same ideas aimed at a student.
In \S2.1 we start with a detailed translation of the definitions by Bourbaki of a dense subspace
and a separation axiom of being Kolmogoroff/$T_0$ and show these definitions implicitly
describe the simplest counterexamples involving spaces consisting of one or two points, and
in fact require orthogonality to these counterexamples. Appendix~\S\ref{app:rtt-examples} and \S\ref{app:rtt-top}
gives more examples of properties defined by iterated orthogonals. Examples include the notions of: compact, discrete, connected, and
totally disconnected spaces, dense image, induced topology, and separation axioms.
Appendix~\S\ref{app:top-notation} introduces a formal syntax and semantics which expresses these properties in several bytes
in both human- and a computer- readable form.
Outside of topology, examples in \S\ref{app:rtt-top} include finite groups being
nilpotent, solvable, torsion-free, $p$-groups, and prime-to-$p$ groups;
injective and projective
modules; injective, surjective.
Compactness is discussed in \S\ref{comp:ult} we reformulate the Bourbaki's definition in terms of convergence of ultrafilters
as an iterated orthogonal of the simplest counterexample. With help of this, we show in \S\ref{comp:m2} that
there is a factorisation system corresponding to Stone-\v Cech compactification, and thus
%its basic properties %of Stone-\v Cech compactification
%can be proven by the Quillen small object argument, and is thus
it is somewhat analogous to Axiom M2 $(cw)(f)$- and $(c)(wf)$- decompositions
required in Quillen model categories.
In \S\ref{ax:chasing} we reformulate the axioms of a topology in a form almost ready to be implemented in a theorem prover based on
diagram chasing with finite preorders. In \S\ref{ax:union:AEEA} we observe that the axiom of topology
saying that an arbitrary union of open subsets is open can be expressed as a formula of form $\forall\exists\varphi\implies\exists\forall\varphi$
and state a speculation Remark~\ref{top:AEEA} that topology is really about permuting quantifiers, a language to talk about dependencies.
In \S3, we ``transcribe'' the informal considerations in [Bourbaki, Introduction]. We ``read off'' from there in \S\ref{def:topoic-top} and \S\ref{met:filt}
that topological and uniform spaces are 2-dimensional simplicial objects
in the same category, the category of filters.
The discussion in \S\ref{exp:limits} of the notion of a limit of a filter $\FFF$ on a topological space $X$
leads to a reformulation
in terms of a lifting property wrt shift (d\'ecalage) simplicial maps forgetting first face and degeneracy maps
$ ( X\times X_{\ttt}, X\times X\times X_\ttt, ...)\lra
(X_\ttt, X\times X_{\ttt}, X\times X\times X_\ttt, ...)$
.
%%the map \catcode`\_=8
%%$$(\FFF,\FFF\times \FFF,...)\lra
%%%(X_{\{X\}},X\times X_{\{X\times X\}}, X\times X\times X_{\{X\times X\times X\}}, ...)\xra{(a,x_1,x_2,x_3,...)}
%%(X, X\times X, X\times X\times X, ...)\xra{(a,x_1,x_2,x_3,...)}
%%(X\times X_{\ttt}, X\times X\times X_\ttt, ...)$$
%%from the object of Cartesian powers of $\FFF$
%%to the shift (d\'ecalage) of the simplicial object $(X_\ttt,X\times X_\ttt,...)$ corresponding to $X$.
%%
%%-- factorising the obvious simplicial map
%%-- $(\FFF,\FFF,...)\lra (X_\ttt, X\times X_{\ttt}, X\times X\times X_\ttt, ...)$
%%-- to the simplicial object $(X_\ttt,X\times X_\ttt,...)$ corresponding to $X$.
%%--
%%-- In these terms,
%%-- taking limit is factorising this map
%%-- via the shift (d\'ecalage)
%%-- $ ( X\times X_{\ttt}, X\times X\times X_\ttt, ...)\lra
%%-- (X_\ttt, X\times X_{\ttt}, X\times X\times X_\ttt, ...)$
%%-- and
%%-- via the full subcategory of setwise Cartesian powers, %, cf.~\ref{exp:limits}.
%%-- namely as
%%-- %%$$
%% \xymatrix{ {} & {{\mathfrak F}} \ar[r]|{\text{id}} \ar@{->}[d] & {{\mathfrak F}} \ar[r]|{\text{id}}\ar[l] \ar[d] & {{\mathfrak F}} \ar[r]|{\text{id}}\ar[l] \ar[d] & {} \ar[l] \\
%%{} &{F_{\mathfrak B} } \ar[r]\ar[d]|{x\mapsto(a,f'(x))} & { {F\times F}_{\mathfrak B} } \ar[r]\ar[l]\ar[d]|{x\mapsto(a,f'(x),f'(x))} & { { F\times F\times F}_{\mathfrak B} } \ar[r]\ar[l]\ar[d]|{x\mapsto(a,f'(x),f'(x),f'(x))} & {} \ar[l] \\
%%{X} \ar[r] &{ {X \times X}_{\ttt} } \ar[r]\ar[l]\ar[d]|{(x_1,x_2)\mapsto x_2} & { {X \times X\times X}_\ttt } \ar[r]\ar[l]\ar[d]|{(x_1,x_2,x_3)\mapsto (x_2,x_3)} & { {X \times X\times X\times X}_\ttt } \ar[r]\ar[l]\ar[d]|{(x_1,x_2,x_3.x_4)\mapsto (x_2,x_3.x_4)} & {} \ar[l] \\
%%{} & {X} \ar[r] &{ {X \times X}_{\ttt} } \ar[r]\ar[l] & { {X \times X\times X}_\ttt } \ar[r]\ar[l] %& { {X \times X\times X\times X}_\ttt } \ar[r]\ar[l]
%%& {} \ar[l]
%%}
%%$$
%%
\def\diag{\mathrm{diag}}
$$
\xymatrix
{ {} & {} \\
{} & {} \\
{} & {\ttt(X)_+}\ar[d]|{pr_{2,3,...}}\\
{E_\diag(F)}\ar@{-->}[ur]\ar[r] & {\ttt(X)}
}
\xymatrix{{}&{}}
\xymatrix
{{...} \ar[d]\ar@/_1pc/[rr]& {...} \ar[d] \ar[dr]|{pr_{2,..}} & {...}\ar[d] \\
{F\times F\times F_{\diag F}}\ar[d]\ar@/_1pc/[rr] \ar@{-->}[ur] & {X\times X\times X_\ttt}\ar[d] \ar[dr]|{pr_{2,3}} & {X\times X\times X_\ttt} \ar[d] \\
{F\times F_{\diag F}}\ar[d]\ar@/_1pc/[rr] \ar@{-->}[ur] & {X\times X_\ttt} \ar[dr]|{pr_2} & {X\times X_\ttt} \ar[d] \\
{F}\ar[rr] \ar@{-->}[ur] & {} & {X}
}
$$
where $F\times ... \times F$ is equipped with the finest filter such that the face
maps (diagonal embeddings) $F\lra {F\times ... \times F}, \ x\mapsto (x,...,x)$ are continuous,
i.e.~a subset of $F\times ... \times F$ is $\diag F$-big iff it contains the image of a $F$-big subset of $F$
under the face map $F\lra {F\times ... \times F}$.
We end the section with a discussion in \S\ref{cwf-decompositions} of path spaces and cylinder objects in the category of topological spaces;
this also leads to constructions reminiscent of the shift (d\'ecalage).
Note that the d\'ecalage of a simplicial set is a model for the path space object of a topological space,
somewhat smaller than the usual model we discuss.\footnote{See~\href{https://ncatlab.org/nlab/show/decalage}{[nlab:d\'ecalage]} for a detailed discussion.}
In \S4, we formulate a number of open questions. Unfortunately, interesting open questions are rather vague and concern
the expressive power and formalisation of the new category theoretic, diagram-chasing way to talk about topology; to what extent the new language helps to avoid irrelevant set-theoretic details and counterexamples.
An important precise open question in this spirit is to define a model structure on the simplicial category of filters
compatible with a model structure on the full subcategory of topological spaces. An application which would be of interest
in geometry is to formulate a clean version of Arzela-Ascoli theorem (cf.~Question~\ref{q:ascoli}), i.e.~a compactness principle for function spaces
of maps between topological, metric and/or measurable spaces.
\subsection{Speculations.}
Does your brain (or your kitten's) have the lifting property (orthogonality), simplicial objects or diagram chasing built-in?
\S2 suggests a broader and more flexible context making
contemplating an experiment possible. Namely, some standard
arguments in point-set topology are computations with
category-theoretic (not always) commutative diagrams of finite categories (which happen to be preorders,
or, equivalently, finite topological spaces) in the same way
that lifting properties define injection and
surjection. In that approach, the lifting property is viewed as a rule to add a new arrow,
a computational recipe to modify diagrams.
Can one find an experiment
to check whether humans {\em subconsciously} use diagram chasing to reason about topology?
Does it appear implicitly in old original papers and books on point-set topology?
Is diagram chasing with preorders too complex to have evolved? Perhaps; but note the self-similarity:
preorders are categories as well, with the property that there is at most
one arrow between any two objects; in fact sometimes these categories are
thought of as $0$-categories. So essentially your computations are in
the category of (finite $0$-) categories.
Is it universal enough? Diagram chasing and point-set topology, arguably
a formalisation of ``nearness'',
is used as a matter of course in many arguments in mathematics.
Finally, isn't it all a bit too obvious? %Does everybody know these properties or finds them obvious?
Curiously, in my experience it's a party topic people often get stuck on.
If asked, few if any can define a surjective or an injective map without words,
by a diagram,
or as a lifting property, even if given the opening sentence of \S\ref{app:sur-and-in} %the previous section
as a hint.
No textbooks seem to bother to mention these reformulations (why?).
An early version of \href{http://mishap.sdf.org/mints/Exercises_de_style_A_homotopy_theory_for_set_theory-I-II-IJM.pdf}{[Gavrilovich, Hasson]} states (*)${}_{\rtt}$ and (**)${}_{\rtt}$ of \S\ref{intro:sur} and \S\ref{app:sur-and-in}
as the simplest examples of lifting properties we were able to think up;
these examples were removed while preparing for publication.
%%%---
%%%---
%%%---
%%%--- No effort has been made to provide a complete bibliography;
%%%--- the author shall happily include any references
%%%--- suggested by readers in the next version [G].
%------------------------
%A useful intuition is to think that the property of being left orthogonal to % left-lifting against
%a class \ensuremath{C} is a kind of negation of the property of being in \ensuremath{C}, and that
%being right orthogonal %right-lifting is another kind of negation.
%%%--- Taking the orthogonal of a property (class) \ensuremath{C} of morphisms
%%%--- is a simple way to define a class of morphisms without the property (excluding
%%%--- non-isomorphisms from \ensuremath{C}), in a way which is useful in a
%%%--- diagram chasing computation.
%%%---
%%%---
%%%---
%%%--- ....We find that taking left or right ortogonal of a class of morhpisms is a
%%%--- particularly common construction, which we think of (and suggest to call) as
%%%--- {\em left or right negation}: ....
%%%---
%%%---
%%%---
%%%---
%%%--- We find that a number of apparently unrelated elementary topological properties
%%%--- >>once translated >to the language of category theory, follow the same pattern and??
%%%--- are obtained by repeatedly passing to the left or right orthogonal $C^l, C^r, C^{lr}, C^{ll}, C^{rl}, C^{rr},...$
%%%--- starting from a very simple class of morphisms, often
%%%--- a single archetypal (counter)example to the property you define.
%%%---
%%%---
%%%--- [??] shows that examples include the notions of: compact, discrete, connected, and
%%%--- totally disconnected spaces, dense image, induced topology, and separation axioms.
%%%---
%%%--- ...Beacuse of this, we suggest to call {left or right orthogonal} as {\em left or right negation}...
%%%---
%%%---
%%%---
%%%---
%%%--- In fact, a number of notions outside of topology may also be defined in this way,
%%%--- for example finite groups being
%%%--- nilpotent, solvable, torsion-free, $p$-groups, and prime-to-$p$ groups;
%%%--- injective and projective
%%%--- modules; injective, surjective,
%%%--- and split homomorphisms (see [???]). Because of this, we think
%%%--- this observation is of independent interest.
%%%---
%%%---
%%%--- ...>>> For us the main contribution of this paper is an attempt to bring attention
%%%--- to .. that an unsophisticade process uncovers category theoretic constructions,
%%%--- and .. taking orgothogonal as negation. ???
%%%---
%%%---
%%%--- Mathematically, we observe the following.
%%%---
%%%--- The categories of topological space and of uniform spaces are,
%%%--- in a natural way, full subcategories of the same larger category, namely the
%%%--- simplicial category of filters, and that this implicit in the definitions of a
%%%--- topological and uniform space. We reformulate the notion of a limit using
%%%--- the shift of simplicial category, and make some suggestions on how to extend
%%%--- the model structure of topological spaces to the ambient category.
%%%--- In [??]
%%%--- we use these embeddings to rewrite the notions of completeness, precompactness, compactness, Cauchy sequence, and equicontinuity
%%%--- in the language of category theory in a rather direct>verbitim? manner.
%%%--- %Our o
%%%---
%%%--- ....[??] shows that the following notions many standard elementary notions
%%%--- of abstract topology can be defined as iterated orthogonals (negation)
%%%--- of the class of morphisms consisting of a single simple morphism of finite topological spaces.
%%%--- Examples in topology include the notions of: compact, discrete, connected, and
%%%--- totally disconnected spaces, dense image, induced topology, and separation axioms.
%%%---
%%%---
%%%--- Examples in algebra include: finite groups being
%%%--- nilpotent, solvable, torsion-free, $p$-groups, and prime-to-$p$ groups;
%%%--- injective and projective
%%%--- modules; injective, surjective,
%%%--- and split homomorphisms.
%%%---
%%%--- %Taking the orthogonal of a class \ensuremath{C} is a simple way to define a class of
%%%--- %morphisms excluding non-isomorphisms from \ensuremath{C}, in a way which is useful in a
%%%--- %diagram chasing computation.
%%%---
%%%---
%%%--- \subsubsection{leftovers of intro}---------------
%%%--- ................
%%%--- We present some naive and technically unsophisticated considerations
%%%--- which show that
%%%--- a number of elementary arguments in topology {\em implicitly} uses logic and reasoning of
%%%--- category theory, ... can be expressed in very concise notation ...
%%%---
%%%--- ... in a sense our goal is (quasi-)historic: we'd like to argue that ideas of category theory
%%%--- were already understood by Hausdorff etc but they had no language (or inclination) to express them. ..
%%%---
%%%--- ... we perform a logical analysis (? analysis of logic ?) of certain arguments
%%%--- and see they correspond to combinatorial(?) manipulations which seem to be part of a logic ..
%%%---
%%%---
%%%--- We give several examples where we ``read off'' category theoretic constructions ...
%%%---
%%%--- ... we believe, if our notation for topological properties properly developed, it can be ``read off''
%%%--- by a machine learning algorithm from the text of say Bourbaki ...
%%%---
%%%--- ... develop a proper notation and a formal system ...
%%%---
%%%--- ... we believe, if our notation for topological properties properly developed, it can be ``read off''
%%%--- by a machine learning algorithm from the text of say Bourbaki ...
%%%---
%%%--- ... develop a proper notation and a formal system ...
%%%---
%%%--- ... nothing new, just notation for what is there already ...
%%%--- ...
%%%---
%%%---
%%%--- {\tt FIXME/TODO: I do not introduce notation for orthogonals. where and how do i do it?
%%%--- ideally i want to give examples and leave full definitions for another text.
%%%--- I do not say much orthogonals in general...
%%%---
%%%--- This is only a draft of the main body of the text; much has to be added.}
%%%---
%%%--- {\tt NIEUW: }.. a naive, almost? textual? analysis of the {\em text} of >[Bourbaki,1st chapter]?
%%%--- uncovers standard construction in category theory...
%%%---
%%%--- ...We show how the language of category theory is implicit in the words and motivation of [Bourbaki, 1st chapter] ...
%%%---
\subsection{Surjection: an example}\label{intro:sur} Let us now explain what we mean by translation.
A map $f:X\lra Y$ is {\em surjective} iff it is left-orthogonal to the simplest non-surjective map $\emptyset\lra\{\bullet\}$, i.e.
$$\!\!\mathrm{(*)_{\rtt}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \emptyset\lra\{\bullet\} \rtt X \xra f Y$$
Recall that for morphisms $f:A\longrightarrow B$, $g:X\longrightarrow Y$ in a category, {\em a morphism $f$ has the left lifting property
wrt a morphism $g$}, {\em $f$ is (left) orthogonal to $g$}, and we write $f
\,\rightthreetimes\, g$ or $A \xra f B \,\rightthreetimes\, X \xra g Y $,
iff for each $i:A\longrightarrow X$, $j:B\longrightarrow Y$ such that $ig=fj$ (``the square commutes''),
there is $j':B\longrightarrow X$ such that $fj'=i$ and $j'g=j$ (``there is a diagonal
making the diagram commute'').
With this definition, $\mathrm{(*)_{\rtt}}$ reads as
\begin{itemize}
\item[$\mathrm{(*)_{\text{words}}}$]
{\em for each map $\{\bullet\}\xra y Y$, i.e. a point $y\in Y$,
there is a map $\{\bullet\}\xra x X$, i.e. a point $x\in X$, such that $f\circ x=y$, i.e. $f(x)=y$}.
\end{itemize}
This is the {\em text} of the usual definition of surjectivity of a function found in an elementary textbook.
Conversely, we can read off $\mathrm{(*)_{\rtt}}$ from the text of the
definition of surjectively, by drawing the commutative diagram as we read
$\mathrm{(*)_{\text{words}}}$.
It is this kind of direct, almost syntactic, relationship between
the usual text and its category theoretic reformulation
we are looking for in this paper.
%We'd like to say that
This is what we mean by saying {\em the reformulation $\mathrm{(*)_{\rtt}}$
is implicit in the text $\mathrm{(*)_{\text{words}}}$}.
For a property (class) \ensuremath{C} of arrows (morphisms) in a category,
define its {\em left} and {\em right orthogonals}, which we also call its
{\em left} and {\em right negation}
$$ C^l := \{ \ensuremath{f} :\text{ for each }g \in C\ \ensuremath{f} \,\rightthreetimes\, \ensuremath{g} \} $$
$$ C^r := \{ \ensuremath{g} :\text{ for each }f \in C\ \ensuremath{f} \,\rightthreetimes\, \ensuremath{g} \} $$
$$ C^{lr}:=(C^l)^r,\, C^{ll}:=(C^l)^l, ... $$
Take $C=\{ \emptyset\longrightarrow \{*\} \}$ in $Top$. A calculation %(see below)
shows
that a few of its iterated negations are meaningful: in $Top$,
$C^r$ is the class of surjections (as we saw earlier), $C^{rr}$ is the class of subsets,
$C^{rl}$ is the class of maps of form $A\longrightarrow A\sqcup D$, $D$ is discrete;
$\{\bullet\}\lra A$ is in $C^{rll}$ iff $A$ is connected;
$Y$ is totally disconnected iff $\{\bullet\}\xra y Y$ is in $C^{rllr}$ for each map $\{\bullet\}\xra y Y$ (or,
in other words, each point $y\in Y$).
% $C^l$ consists of maps $f:A\longrightarrow B$ such that either $A=B=\emptyset$
% or $A\neq \emptyset$, \ensuremath{B} arbitrary.
%Somewhat surprisingly, in both Sets and Top,
$C^{l}$
is the class of maps $A\longrightarrow B$ such that either $A\neq\emptyset$ or $A=B=\emptyset$.
$C^{ll}$
is the class of isomorphisms.
$C^{lr}$ is the class of maps $\emptyset\lra B$, $B$ is arbitrary
$C^{lrl}$ is the class of maps which admit a section.
$C^{lll}=C^{llr}=..$ is the class of all maps.
%more generally, for any $lr$-string $w=w_1...w_i$,$i>0$,
%of odd length,
%$C^{lw_1,...,w_{2i+1}}$ is the class of isomorphisms,
%and $C^{lw_1,...,w_{2i+2}}$ is the class of all maps.
Thus we see that already in this simplest case, taking iterated orthogonals (negation) produces
several notions from a textbook,
namely surjective, subset, discrete, connected, non-empty, and totally disconnected.
\subsection{Intuition/Yoga of orthogonality.}\label{yoga:ort} We suggest the following intuition/yoga is helpful.\footnote
{We were unable to find literature which explicitly describes this intuition, and will be thankful for any references which either discuss this intuition or list potential (counter)examples.
}
\begin{itemize}
\item taking iterated orthogonals (negation) is a cheap way to automatically ``generate'' interesting notions;
a number of standard textbook notions are obtained in this way.
We saw that taking iterated negations of the simplest map of topological spaces, $\{\}\lra\{\bullet\}$,
generates $5$ classes worthy of being defined in a first year course of topology (surjective, subset, discrete, connected,
non-empty and totally disconnected).
\item it helps to think of {\em orthogonality as a category-theoretic (substitute for) negation};
taking orthogonal is perhaps the simplest way to define a class of morphisms
without a property in a manner useful for a diagram chasing calculation.
\item
often a morphism-counterexample can be ``read off'' from the text of the definition of an elementary textbook property,
and the property can be concisely reformulated as the orthogonal of the class
consisting of that counterexample.
\end{itemize}
\subsection{Intuition/Yoga of transcription.} We suggest the following intuition/yoga is helpful.
\begin{itemize}
\item
``transcribing'' the usual text of mathematical definitions and arguments
by means of notation extensively using arrows sometimes makes it possible to recognise familiar patterns of
standard category-theoretic constructions and diagram chasing arguments.
\item orthogonality of morphisms often appears in this way, and so do simplicial objects
\item from the text of the definition of a topological property sometimes it is possible to ``read off''
a definition of a topology or a filter or a continuous function; it is worthwhile to try to interpret
``for each open subsets there exists ...'' as a requirement that some function is continuous
\ei
\section{Examples of translation. Orthogonality as negation.}
\subsection{Dense subspaces and Kolmogoroff $T_0$ spaces.}
We shall now transcribe the definitions of {\em dense} and {\em Kolmogoroff $T_0$} spaces.
\subsubsection{``$A$ is a dense subset of $X$.''}
By definition \href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki, I\S1.6, Def.12]},
%DEFINITION 12. A
\newline\noindent\includegraphics[width=\linewidth]{Bourbaki-Dense-subset-def.png}
%%+ \begin{quote}
%%+ {\sf {\sc Definition 12.} A subset
%%+ $A$ of a topological space $B$ is said to be dense in $B$
%%+ (or simply dense, if there is no ambiguity about $B$) if $\bar A = B$, i.e.
%%+ {\em if every non-empty open set $U$ of $X$ meets $A$}.}
%%+ \end{quote}
Let us transcribe this by means of the language of arrows.
{\sf A subset $A$ of a topological space $X$} is an arrow $A\lra X$.
(Note there is an alternative translation analogous to the used in the next item).
An {\sf open subset $U$ of $X$} %An {\em open subset} of $B$
is an arrow $X\lra \{ U \ra U' \}$ ; here $\{ U \ra U' \}$ denotes the topological space
consisting of one open point $U$ and one closed point $U'$; by the arrow $\ra$ we mean that
that $U'\in cl(U)$.
{\sf Non-empty}: a subset $U$ of $X$ is {\em empty} iff
the arrow $X\lra \{ U \ra U' \}$ factors as $X\lra \{ U' \}\lra \{ U \ra U' \}$ ;
here the map $ \{ U' \}\lra \{ U \ra U' \}$ is the obvious map sending $U'$ to $U'$.
{\sf set $U$ of $X$ meets $A$}: $U\cap A =\emptyset$ iff the arrow $A\lra X\lra \{ U \ra U' \}$ factors as
$A\lra \{ U' \} \lra \{ U \ra U' \}$.
Collecting above (Figure 1c), we see that
a map $A\xra f X $ has dense image iff
$$ A \xra f X \rtt \{ U' \}\lra \{ U \ra U' \}$$
Note a little miracle:
$\{ U' \}\lra \{ U \ra U' \}$ is the simplest map whose image isn't dense.
We'll see it happen again.
%and thus the property of ``dense image'' is defined as left negation of (of
%the property of being)
%the simplest map without the property>>which does not have dense image.??.
\subsubsection{Kolmogoroff spaces, axiom $T_0$.} By definition \href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki,I\S1, Ex.2b; p.117/122]},
\newline\noindent\includegraphics[width=\linewidth]{Bourbaki-Kolmogoroff.png}
%%+ \begin{quote}
%%+ {\sf b) A topological space is said to be a {\em Kolmogoroff} space if it satisfies the fol-
%%+ lowing condition: given any two distinct points $x$, $x'$ of $X$, there is a
%%+ neighbourhood of one of these points which does not contain the other.}
%%+ \end{quote}
%%+ % a space $X$ satisfies $T_1$ iff
%for each two points $x\neq y\in X$ there is an open subset $U$ containing $x$ but not $y$.
Let us transcribe this. {\sf given any two ...~points $x$, $x'$ of $X$}: given a map $\{x,x'\}\xra f X$.
{\sf two {\em distinct} points}: the map $\{x,x'\}\xra f X$ does not factor through a single point,
%which we choose to denote
i.e. $\{x,x'\}\lra X$ does not factor as $\{x,x'\}\lra \{ x=x' \} \lra X$.
%``an open subset $U$ containing $x$ but not $y$'':
The negation of the sentence {\sf there is a neighbourhood which does not contain the other} defines a topology on the set $\{x,x'\}$: indeed,
the antidiscrete topology on the set $\{x,x'\}$ is the only topology with %defined by
the property that
%{\em each neighbourhood of one of these points
%necessarily contains the other};
{\sf there is [no] neighbourhood of one of these points which does not contain the other}.
Let us denote this space as $\{x \llrra x'\}$.
Now we note that the text implicitly defines
the space $\{x \llrra x'\}$, and the only way to use it is to consider
a map $\{x \llrra x'\}\xra f X$ instead of the map $\{x,x'\}\xra f X$.
%the preimage of an open subset $U$ under the map $\{x,y\}\xra f X$ contains $x$ but not $y$.
%Quantification over preimages of open subsets appears in the definition of continuity, so lets try to use that.
%The property fails iff for each open subset $U$ of $X$, $x\in f\inv(U)$ implies $y\in f\inv(U)$.
%Use this to define a topology on the set $\{x,y\}$: a subset $V\subseteq \{x,y\}$ is open iff $x\in V$ implies $y\in V$, i.e.
%iff $V\in \{ \emptyset, \{y\}, \{x,y\}\}$, i.e. $y$ is open and $x$ is closed.
%Denote the space equipped with this topology as $\{y\ra x \}$. Now it says: each map
%$\{y\ra x \} \xra f X$ factors as $\{x,y\}\lra \{ x=y \} \lra X$ (see Figure 1b).
%Note this is the same lifting property? orthogonality? diagram as in Figure 1a.
Collecting above (see Figure~1d), we see that {\em a topological space $X$ is said to be a {\em Kolmogoroff} space
iff any map $\{x \llrra x'\}\xra f X$ factors as $\{x \llrra x'\}\lra \{ x=x' \} \lra X$.}
Note another little miracle: it also reduces to orthogonality of morphisms
$$ \{x \llrra x'\}\lra \{ x=x' \} \rtt X\lra \{ x=x' \} $$
and $\{ x\llrra x' \}$ is the simplest non-Kolmogoroff space.
%\setcounter{figure}{1}
\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} }}
\begin{figure}
\begin{center}
\large
$ (a)\ \xymatrix{ A \ar[r]^{i} \ar@{->}[d]|f & X \ar[d]|g \\ B \ar[r]|-{j} \ar@{-->}[ur]|{{\tilde j}}& Y }$% \
%$\rrt ABXY$\ \ \ \
$(b)\ \rrt {\{\}} {} {\{\bullet\}} X {\therefore(surj)} Y $%\
$(c)\ \rrt {A} {\therefore(dense)} {B} {\{U'\}} {} {\{U\ra U'\} } $%\
$(d)\ \rrt {\{ x\llrra x' \}} {} { \{ x=x' \}} X {\therefore(T_0)} {\{ x=x' \}} $%\
%$(b)\ \rrt {\{\}} {} {\{\bullet\}} X {\therefore(surj)} Y $%\
%$(c)\ \rrt {\{\bullet,\bullet\}} {} {\{\bullet\}} X {\therefore(inj)} Y $%\ \
%$(d)\ \rrt X {\therefore(inj)} {Y} {\{x,y\}} {} {\{x=y\}}$\
\end{center}
\caption{\label{fig1}%\normalsize
Lifting properties. Dots $\therefore$ indicate free variables and what property of these variables is being defined;
%, i.e.~a property of what is being defined and how is it to be labelled
in a diagram chasing calculation, "$\therefore(dense)$" reads as:
given a (valid) diagram, add label $(dense)$ to the corresponding arrow.\newline
(a) The definition of a lifting property $f\rtt g$: for each $i:A\lra X$ and $j:B\lra Y$
making the square commutative, i.e.~$f\circ j=i\circ g$, there is a diagonal arrow $\tilde j:B\lra X$ making the total diagram
$A\xra f B\xra {\tilde j} X\xra g Y, A\xra i X, B\xra j Y$ commutative, i.e.~$f\circ \tilde j=i$ and $\tilde j\circ g=j$.
(b) $X\lra Y$ is surjective %\newline
(c) the image of $A\lra B$ is dense in $B$
(d) $X$ is Kolmogoroff/$T_0$
% (c) $X\lra Y$ is injective; $X\lra Y$ is an epicmorphism if we forget %never use
%that $\{\bullet\}$ denotes a singleton (rather than an arbitrary object
%and thus $\{\bullet,\bullet\}\lra\{\bullet\}$ denotes an arbitrary morphism $Z\sqcup Z\xra{(id,id)} Z$)\newline
% (d) $X\lra Y$ is injective, in the category of Sets; $\pi_0(X)\lra\pi_0(Y)$ is injective,
% when the diagram is interpreted in the category
%of topological spaces.
}
\end{figure}
\subsubsection{Finite topological spaces as categories.}
Our notation $\{ U' \}\lra \{ U \ra U' \}$ and $\{x \llrra x'\}\lra \{ x=x'
\}$ suggests that {\em we reformulated the two topological properties of being dense and Kolmogoroff
in terms of diagram chasing in (finite) categories}. And indeed, we may think
of finite topological spaces as categories and of continuous maps between them as {\em functors},
as follows; see Appendix~\ref{app:top-notation} for details and a definition of
our notation for finite topological spaces and maps between them.
A {\em topological space} comes with a {\em specialisation preorder} on its points: for
points $x,y \in X$, $x \leq y$ iff $y \in cl x$ ($y$ is in the {\em topological closure} of $x$).
The resulting {\em preordered set} may be regarded as a {\em category} whose
{\em objects} are the points of ${X}$ and where there is a unique {\em morphism} $x{\ra}y$ iff $y \in cl x$.
For a {\em finite topological space} $X$, the specialisation preorder or
equivalently the corresponding category uniquely determines the space: a {\em
subset} of ${X}$ is {\em closed} iff it is
{\em downward closed}, or equivalently,
it is a subcategory such that there are no morphisms going outside the subcategory.
The monotone maps (i.e. {\em functors}) are the {\em continuous maps} for this topology.
We denote a finite topological space by a list of the arrows (morphisms) in the
corresponding category; '$\leftrightarrow $' denotes an {\em isomorphism} and
'$=$' denotes the {\em identity morphism}. An arrow between two such lists
denotes a {\em continuous map} (a functor) which sends each point to the
correspondingly labelled point, but possibly turning some morphisms into
identity morphisms, thus gluing some points.
\subsection{\label{comp:ult}Compactness via ultrafilters.}
We try to interpret the definition of compactness in \href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki,I\S9.1, Def.1(C$'$)]}
in terms of arrows, or rather we try to rewrite it using the arrow notation
as much as possible. Doing so we shall see that
this definition, in appropriate notation, condenses to
{\em a Hausdorff space $K$ is quasi-compact iff $K\lra\{\bullet\}$ is in
$$
((\{ \{o\}\lra \{o\ra c\}\}^{r})_{<5})^{lr},$$}
and that the latter expression almost appears in [Bourbaki, I\S10.2,Thm.1d]
as a characterisation of the class of proper maps.
\subsubsection{Reading the definition of quasi-compactness.} We read the definition of quasi-compactness [Bourbaki,I\S9.1, Def.I];
we do not know how to read (C) and therefore we read its reformulation (C$'$).
\newline\noindent\includegraphics[width=\linewidth]{Bourbaki-compact-def-full-Cprime.png}
%\newline\noindent\includegraphics[width=\linewidth]{Bourbaki-compact-Def-I-1.png}
%\newline\noindent\includegraphics[width=\linewidth]{Bourbaki-compact-Def-I-Cprime.png}
%%%
%%%\begin{quote}\tt
%%%{\sc DEFINITION I}. {\em A topological space X is said to be quasi-compact if it satisfies
%%%the following axiom:
%%%%%+ (C) Every .filter on X has at least one cluster point.
%%%%%+ A topological space is said to be compact if it is quasi-compact and Hausdorff.
%%%%%+ It follows immediately from this axiom that if f is a mapping of a set Z
%%%%%+ into a quasi-compact space X, and
%%%%%+ is any filter on Z, then f has
%%%%%+ at least one cluster point with respect to
%%%%%+ . In particular, every sequence
%%%%%+ of points of a quasi-compact space has at least one cluster point; but
%%%%%+ this condition is not equivalent to (C) (Exercise 1 I).
%%%%%+ We give three axioms each of which is equivalent to axiom (C):
%%%$(\rm C')$ Every ultrafilter on $X$ is convergent.
%%%}\end{quote}
%%%%%%
%%%%%%\begin{quote} {\sc DEFINITION I}. Let $X$ be a topological space and $\UUU$ a filter on $X$.
%%%%%%A point
%%%%%%$x \in X$ is said to be a limit point (or {\em simply} a limit) of
%%%%%%$\UUU$, if $\UUU$
%%%%%% is finer than the
%%%%%%neighbourhood filter
%%%%%%$\mathfrak B (x)$ of $x$; $\mathfrak F$
%%%%%% is also said to converge (or {\em to be} convergent)
%%%%%%to $x$. The point $x$ is said to be a limit of a filter base
%%%%%% on $X$, and
%%%%%% is
%%%%%%said to converge to $x$, if the filter whose base is
%%%%%% converges to $x$.
%%%%%%\end{quote}
A space $K$ is {\em quasi-compact} iff each ultrafilter $\UUU$ on the set of points of
$K$ converges,
i.e. for each ultrafilter $\UUU$ on the set of points of
$K$ there is a point $x \in K$ such that each open neighbourhood of $x$ is $\UUU$-big.
This contains a quantification over open subsets; this suggests to us that
we should try to extract a definition of topology from the text and to interpret
the requirement as continuity of a certain map.
{\sf each open neighbourhood of $x$ is $\UUU$-big} suggest we define a topology such that
an open subset is an $\UUU$-big open neighbourhood of some $x\in K$. This defines
a topology on \def\oo{\text{``x''}} $K\sqcup \{\oo\}$:
$$\{\,U \,:\, U\subset K\text{ is open}\}\cup \{\,U\cup\{\oo\} \,:\, U\subset K\text{ is open and }\UUU\text{-big}\}$$
%We do so by essentially turning this property into a definition of a topology.
%Preimages of open neighbourhood of a point constitute open neighbourhoods of its preimage, hence we consider
%a map $K\sqcup \{\oo\} \xra {f_x} K$, $f_x(\oo)=x$, and the topology on $K\sqcup \{\oo\}$ is such that
%(a) a subset containing $\oo$ is open iff it is of form $\{\oo\}\sqcup U$ where $U\in \UUU$ is an open subset of $K$
%(b) a subset not containing $\oo$, i.e. a subset of $K$, is open iff it is open in $K$.
%Item (b) ensures that continuity of
%$$f_x:K\sqcup\{\oo\}\lra K,\ \oo\mapsto x ;\ {\text{ for }}y\in K\ y\mapsto y$$ does not
%place unnecessary restrictions on $f_x$).
Denote the set equipped with this topology
by $K\sqcup_{\UUU} \{\oo\}$. (Note [Bourbaki, I\S6.5, Definition 5, Example] define this space.)
Thus, in terms of arrows the definition becomes (see Figure 2a):
$K$ is quasi-compact iff
the identity map $K \xra {\id} K$
factors as $$K\lra K\sqcup_{\UUU} \{\oo\} \lra K$$ for each ultrafilter $\UUU$ on the set of points of $K$.
Now note that Figure 2a is a particular case of orthogonality $K\lra K\sqcup_{\UUU} \{\oo\} \rtt K\lra\{\bullet\}$,
see Figure 2b where the map $K\lra K$ is arbitrary. Using orthogonals (negation), we express this by saying
that $K\lra K\sqcup_{\UUU} \{\oo\} \in \{ K\lra \{\bullet\}\}^l$.
As usual, we are tempted to define compactness as an orthogonal (negation) of a class (property) of morphisms,
and therefore we check that all maps of form $A\lra A\sqcup_{\UUU}\{\bullet\}$ lie
in this orthogonal $\{ K\lra \{\bullet\}\}^l$. Conversely, this also means that $K\lra \{\bullet\}$, for $K$ quasi-compact,
lies in the right orthogonal (negation) $\{\,A\lra A\sqcup_{\UUU}\{\bullet\} \,:\, \UUU\text{ is an ultrafilter on a space }A\,\}^r$.
Summing up, we read Definition I as
\begin{quote}
%{\sc DEFINITION I}. {\em A topological space $X$ is said to be quasi-compact iff\\
%\newline
\noindent\includegraphics[width=\linewidth]{Bourbaki-compact-Def-I-1.png}
$(\rm C')_{\rtt}$ $A\lra A\sqcup_{\UUU}\{\bullet\} \rtt X\lra \{\bullet\}$ for each ultrafilter $\UUU$ on each space $A$
%$(\rm C')_{\rtt}$ $X\lra \{\bullet\}\in \{\,A\lra A\sqcup_{\UUU}\{\bullet\} \,:\, \UUU\text{ is an ultrafilter on a space }A\,\}^r$
\end{quote}
Note that there is another, more direct, way to read off this lifting property $(\rm C')_{\rtt}$
from a remark in the proof of $(C)\implies (C')$:
\vskip3pt%\newline
\noindent\includegraphics[width=\linewidth]{Bourbaki-compact-def-as-lifting.png}
%
%{\sf It follows immediately from [(C$'$)] that if $f$ is a mapping of a set $Z$
%into a quasi-compact space $X$, and [$\mathfrak U$]
% is any [ultra]filter on $Z$, then $f$ has
%at least one [limit] point with respect to [$\mathfrak U$].}
%In particular, every sequence
%of points of a quasi-compact space has at least one cluster point; but
%this condition is not equivalent to (C) (Exercise 1 I).
%
In terms of arrows, this reformulation is {\em precisely} the lifting property
$$Z\lra Z\sqcup_{\UUU} \{\oo\} \rtt X\lra\{\bullet\}$$
We'd like to view the fact that Bourbaki chooses to formulate explicitly {\em precisely} a lifting property
immediately following a key definition
as evidence that Bourbaki is implicitly doing category theoretic reasoning.
\subsubsection{Proper maps.} If we were to think that
\href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki, General Topology]} does implicitly uses category theoretic reasoning and orthogonality,
we'd hope to find there the definition of the class
$$\{\,A\lra A\sqcup_{\UUU}\{\bullet\} \,:\, \UUU\text{ is an ultrafilter on a space }A\,\}^r$$
And indeed, this is how Bourbaki characterises the class of proper maps in
\href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki, General Topology, I\S10.2,Th.1(d)]}
(cf.~Figure~2d), almost exactly.
We see this as evidence that Bourbaki does indeed use category theoretic reasoning, or perhaps
as an explanation of what do we mean by saying so.
%Astonishing, this class defiend in bourbaki; so ...
%If we were to think that
%\href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki, General Topology]} implicitly uses category theoretic reasoning,
%as this is exactly how Bourbaki defines the class of proper maps.
%hope to find this class defined in almost these terms.
%And indeed, we do find that %it defined in o the notion of (and almost to the definition) of a proper map by Bourbaki
%\href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki, General Topology, I\S10.2,Th.1(d)]} (cf.~Figure~2d).
Note we might have started our translation with this characterisation of proper maps
in terms of ultrafilters \href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki, General Topology, I\S10.2,Th.1(d)]}, and
we'd then arrive at Figure~2d directly.
\newline\noindent \includegraphics[width=\linewidth]{Bourbaki-proper-maps-thm.png}
However, this reformulation is unsatisfactory for us:
%it mentions ultrafilters which we have little grasp of.
it uses non-elementary, infinitary, set-theoretic
notion of ultrafilters which we do not know how to manipulate category-theoretically..
We'd like to have a definition which relies on maps between finite spaces.
An argument similar to a linear algebra about dual vector spaces gives the following.
For any class $C$ of maps we have that $C^l=C^{lrl}$ and $C^r=C^{rlr}$
and $C_1\subset C_2$ implies $C_1^l\supset C_2^l$ and $C_1^r \supset C_2^r$. This implies
$P^{lr}\subset C^{rlr}=C^{r}$ whenever $P\subset C^r$.
Take $P$ to be some class of proper maps between finite spaces. By above we see that
$P^{lr}$ is a subclass of the class of proper maps. We want to take $P$ to be
large enough so that $P^{lr}$ is the whole class of proper maps.
And indeed, we find that a classical theorem in general topology
tells us we can do so, at least if we only care about spaces satisfying
separation axioms. Moreover, we will see it is enough to take $P$ to consist of
the following
maps between spaces of size at most 3:
\begin{center}$
{ \{\boldsymbol{B_1}\leftarrow O\rightarrow \boldsymbol{B_2}\} } \lra {\{\bullet\} } \ \ \ \ \ \ \ \ \
\{U\} \lra { \{ U \ra U' \} } $ \\
$
{ \{x\llrra y\} } \lra \{x=y\} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
{ \{o \ra c \} } \lra \{o=c\}
$
\end{center}
\subsubsection{Reducing to finite spaces.}
Now we are back translating; we ignore the considerations of the previous subsubsection
which give us a rather good idea of what we would get as the result of translation.
Reduction to finite spaces is provided by Smirnov-Vulikh-Taimanov theorem in the form by
[Engelking, 3.2.1,p.136] (``compact'' below stands for ``compact Hausdorff''):\vskip3pt%\newline
\noindent \includegraphics[width=\linewidth]{Engelking-taimanov.png}
%\begin{quote} 3.2.1. THEOREM. Let $A$ be a dense subspace of a topological space \ensuremath{X} and \ensuremath{f} a continuous
%mapping of $A$ to a compact space $Y$. The mapping \ensuremath{f} has a continuous extension over \ensuremath{X} if and
%only if for every pair $B_1,B_2$ of disjoint closed subsets of \ensuremath{Y} the inverse images $f^{-1}(B_1)$
%and $f^{-1}(B_2)$ have disjoint closures in the space $X$.
%\end{quote}
Let us transcribe this. We are given {\sf a dense subspace $A\xra i X$ of a topological space $X$} and {\sf a continuous mapping $A\xra f Y$ of $A$ to a [Hausdorff] compact space $Y$}.
{\sf The mapping \ensuremath{f} has a continuous extension over \ensuremath{X}}
iff the arrow $A\xra f Y$ factors via $A\xra i X$ (cf.~Figure 2f).
A {\sf pair $\boldsymbol{B_1}$, $\boldsymbol{B_2}$ of disjoint closed subsets of \ensuremath{Y}} is an arrow
$Y\lra \{\boldsymbol{B_1}\leftarrow O \ra \boldsymbol{B_2}\}$
where $\{\boldsymbol{B_1}\leftarrow O \ra \boldsymbol{B_2}\}$ is the space with
one open point denoted by $O$ and two closed points
denoted by $\boldsymbol{B_1}$ and $\boldsymbol{B_2}$.
{\sf the inverse images $\boldsymbol{f^{-1}(B_1)}$
and $\boldsymbol{f^{-1}(B_2)}$ have disjoint closures in the space $X$} says
the arrow $A\xra f Y \lra \{\boldsymbol{B_1}\leftarrow O \ra \boldsymbol{B_2}\}$ factors as $A \xra i X
\lra \{\boldsymbol{B_1}\leftarrow O \ra \boldsymbol{B_2}\}$ (cf.~Figure 2g).
Now we need to define the class of dense subspaces.
A dense subspace is an injective map with dense image
such that the topology on the domain is induced from the target.
This suggests we try to define this class by
%We do so by
taking left negations (orthogonals) of the simplest archetypal examples of
maps with non-dense image, a non-injective map, and a map
such that the topology on the domain is not induced from the target.
%%%''''''''Note that the intuition of thinking of orthogonality of morphisms
%%%as (and terminology of referrering to) as negations helps: with this, items 3.2.1(i-iii) say
%%%that you obtain the notions of being dense, injective, induced topology by {\em negating}
%%% the simplest archetypal example of a map
%%%with non-dense image, non-injective map, a map where topology is not induced.
%%%
\begin{quote} 3.2.1. THEOREM.
Let $Y$ be Hausdorff compact and let $A\xra i X$ satisfy (cf.~Figure 2(ijk))
\begin{enumerate}
\item[(i)] (dense) $A\xra i X \rtt \{U\} \lra { \{ U \ra U' \} } $
\item[(ii)] (injective) $A\xra i X \rtt { \{x\llrra y\} } \lra \{x=y\}$
\item[(iii)] (induced topology) $A\xra i X \rtt { \{o \ra c \} } \lra \{o=c\} $
\end{enumerate}
Then the properties of $A\xra f Y$ defined by Figure 2(f) and Figure 2(g) are equivalent.
\end{quote}
This implies that, for Hausdorff compact $Y$,
items 3.2.1(i-iii) and $ A\xra i X \rtt \{\boldsymbol{B_1}\leftarrow O \ra \boldsymbol{B_2}\}\lra \{\boldsymbol{B_1} = O= \boldsymbol{B_2}\}
$ imply that $ A\xra i X \rtt Y\lra \{\bullet\}$.
Further, note that if $X=A\sqcup\{\oo\}$ is obtained from $A$ by adjoining a single closed non-open point, then
$$ A\xra i X \rtt \{\boldsymbol{B_1}\leftarrow O \ra \boldsymbol{B_2}\}\lra \{\boldsymbol{B_1} = O= \boldsymbol{B_2}\}$$
iff there exists an ultrafilter $\mathfrak U$ such that $A \xra i X$ is of form $A \lra A\sqcup_{\mathfrak U}\{\oo\}$.
This implies that maps of form $A \lra A\sqcup_{\mathfrak U}\{\oo\}$ are in $P^l$ and, finally,
that a Hausdorff space $K$ is quasi-compact iff $K\lra\{\bullet\}$ is in $P^{lr}$ where
$P$ consists of%is
\begin{center}$
{ \{\boldsymbol{B_1}\leftarrow O\rightarrow \boldsymbol{B_2}\} } \lra {\{\bullet\} } \ \ \ \ \ \ \ \ \
\{U\} \lra { \{ U \ra U' \} } $ \\
$
{ \{x\llrra y\} } \lra \{x=y\} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
{ \{o \ra c \} } \lra \{o=c\}
$
\end{center}
%%$$
%%\{\
%%{ \{\boldsymbol{B_1}\leftarrow O\rightarrow \boldsymbol{B_2}\} } \lra {\{\bullet\} } \ ; \
%% \{U\} \lra { \{ U \ra U' \} } \ ; \
%%{ \{x\llrra y\} } \lra \{x=y\} \ ; \
%%{ \{o \ra c \} } \lra \{o=c\} \ \}
%%$$
%%
\subsubsection{The simplest counterexample negated three times.}
Note that all maps between finite spaces mentioned in the preceeding subsubsection %involved in Figure 2(i-k)
are closed, hence proper by \href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki, I\S10.2,Thm.1b]}.
%A map is called closed iff the image of a closed space is closed, and, for finite spaces, it is enough to require
%that the image of a closure of a point is the closure of the image of the point.
A verification shows that, for $Y$ and $Z$ finite, the map $Y\xra g Z $ is closed iff
$$\{o\}\lra \{o \ra c \} \rtt Y\xra g Z$$
Denote by %In notation, we see that $C\subset (\{ \{o\}\lra \{o\ra c\}\}^{r})_{<5}$ where by
$(\{ \{o\}\lra \{o\ra c\}\}^{r})_{<5}$ the subclass of $\{ \{o\}\lra \{o\ra c\}\}^{r}$
consisting of maps between spaces of size at most $4$.
Considerations above could be summarized by:
\begin{enonce*}{Claim} In the category of topological spaces,
\begin{itemize}
\item
a Hausdorff space $K$ is quasi-compact iff
\begin{itemize}
\item[] $K\lra\{\bullet\}$ is in $
((\{ \{o\}\lra \{o\ra c\}\}^{r})_{<5})^{lr}$.
\end{itemize}
\item every map in $((\{ \{o\}\lra \{o\ra c\}\}^{r})_{<5})^{lr}$ is proper
%\item $((\{ \{o\}\lra \{o\ra c\}\}^{r})_{<5})^{lr}$ is a subclass of the class of proper maps.
\end{itemize}
\end{enonce*}
And we conjecture that the latter is in fact the class of all proper maps.
\begin{enonce*}{Conjecture} In the category of topological spaces, the following orthogonal defines the class of proper maps:
$$((\{ \{o\}\lra \{o\ra c\}\}^{r})_{<5})^{lr}$$
\end{enonce*}
%note that all maps between finite spaces mentioned in the preceeding subsubsection %involved in Figure 2(i-k)
%for a map between finite spaces, the notions of proper and closed are equivalent [Bourbaki, I\S10.2,Thm.1b].
%and that maps of form $A \lra A\cup_{\mathfrak U} \{\oo\}$ are as in the assumption of the theorem above.
%
%
%Again, this is reminiscent of the lifting property
%$$ A \lra X \rtt \{\boldsymbol{B_1}\leftarrow O \ra \boldsymbol{B_2}\}\lra \{\bullet\}$$
%...and indeed, being a dense subset can also be expressed in terms of lifting propertings (see Figure 2(i-k)). ...
%%%....
%%%This implies that a Hausdorff space $K$ is quasi-compact iff $K\lra\{\bullet\}$ is in $
%%%\{\ ((\{ \{o\}\lra \{o\ra c\}\}^{r})_{<5})^{lr}$.
%%%....
%%%
%%%Finally, considerations above condence the conjecture that the following
%%%is the class of proper maps:
%%% $$ ((\{ \{o\}\lra \{o\ra c\}\}^{r})_{<5})^{lr}
%%% $$
%%%.....
%%%
%%+ bb
%%+ A map $X\xra g Y $ is proper iff
%%+ \bi
%%+ \iii[] $A\lra A\cup_{\FF} \{\oo\} \rtt X\xra g Y $ for each space $A$ and each ultrafilter $\FF$ on the set of points of $A$
%%+ \ei
%%+
%%+ ....{\tt ASSAF: maybe state the below here as well... }
%%+ the following are equivalent;
%%+ \bi
%%+ \iii[] a space $K$ is compact
%%+ \iii[] $K\lra K\cup_{\FF} \{\oo\} \rtt K\lra \{\bullet\} $ for each ultrafilter $\FF$ on the set of points of $K$
%%+ \iii[] $A\lra A\cup_{\FF} \{\oo\} \rtt K\lra \{\bullet\} $ for each space $A$ and each ultrafilter $\FF$ on the set of points of $A$
%%+ \ei
%%+
%%+ In the next section we shall introduce notation that condenses the above to
%%+ $$ ((\{ \{o\}\lra \{o\ra c\}\}^{r})_{<5})^{lr}
%%+ $$
%%+
\begin{figure}
\small
\begin{center}
$ (a)\ \xymatrix{ K \ar[r]|{\text{id}} \ar@{->}[d] & K \ar[d] \\ { K\cup_{\FF}\{\text{``x''}\} } \ar[r] \ar@{-->}[ur]|{{\text{``x''}\mapsto x}}& \{\bullet\} }$% \
$\ \ \ \ (b)\ \xymatrix{ K \ar[r] \ar@{->}[d] & K \ar[d] \\ { K\cup_{\FF}\{\text{``x''}\} } \ar[r] \ar@{-->}[ur] & \{\bullet\} }$% \
$\ \ \ \ (c)\ \xymatrix{ A \ar[r] \ar@{->}[d] & A \ar[d] \\ { A\cup_{\FF}\{\text{``x''}\} } \ar[r] \ar@{-->}[ur] & \{\bullet\} }$% \
$ (d)\ \xymatrix{ X \ar[r]|{\text{id}} \ar@{->}[d] & X \ar[d]|{f} \\ { X\cup_{\mathfrak U}\{\text{``x''}\} } \ar[r] \ar@{-->}[ur] & Y }$% \
%$(d)\ \rrt X {\therefore(inj)} {Y} {\{x,y\}} {} {\{x=y\}}$\
$\ \ \ \ \ (e)\ \xymatrix{ A \ar[r] \ar@{->}[d] & X \ar[d]|{g} \\ { A\cup_{\mathfrak U}\{\text{``x''}\} } \ar[r] \ar@{-->}[ur] & Y }$% \
$ (f)\ \xymatrix{ A \ar[r]|f \ar@{->}[d] & Y \ar[d]|{g} \\ { X } \ar[r] \ar@{-->}[ur] & \{\bullet\} }$% \
$\ \ \ \ \ (g)\ \xymatrix{ A \ar[r]|f \ar@{->}[d] & Y \ar@{->}[r] & { \{\boldsymbol{B_1}\leftarrow O\rightarrow \boldsymbol{B_2}\} } \\ { X } \ar@{-->}[urr] & { } & { } }$% \
$\ \ \ \ (h)\ \xymatrix{ A \ar[r] \ar@{->}[d] & { \{\boldsymbol{B_1}\leftarrow O\rightarrow \boldsymbol{B_2}\} } \ar[d] \\ { X } \ar[r] \ar@{-->}[ur] & {\{\bullet\} } }$% \
$ (i)\ \ \xymatrix{ A \ar[r] \ar@{->}[d] & { \{U\} } \ar[d] \\ { X } \ar[r] \ar@{-->}[ur] & { \{ U \ra U' \} } }$% \
$\ \ \ \ \ \ (j)\ \ \xymatrix{ A \ar[r] \ar@{->}[d] & { \{x\llrra y\} } \ar[d] \\ { X } \ar[r] \ar@{-->}[ur] & \{x=y\} }$% \
$\ \ \ \ \ \ (k)\ \ \xymatrix{ A \ar[r] \ar@{->}[d] & { \{o \ra c \} } \ar[d] \\ { X } \ar[r] \ar@{-->}[ur] & \{o=c\} }$% \
$ (l)\ \xymatrix{ {\{o\} } \ar[r] \ar@{->}[d] & X \ar[d] \\ { \{o \ra c \} } \ar[r] \ar@{-->}[ur] & { Y } }$% \
%% $ (a)\ \xymatrix{ K \ar[r]^{\text{id}} \ar@{->}[d] & K \ar[d] \\ { K\cup_{\FF}\{\text{``x''}\} } \ar[r] \ar@{-->}[ur]^{{\text{``x''}\mapsto x}}& \{\bullet\} }$% \
%% $ (b)\ \xymatrix{ K \ar[r] \ar@{->}[d] & K \ar[d] \\ { K\cup_{\FF}\{\text{``x''}\} } \ar[r] \ar@{-->}[ur] & \{\bullet\} }$% \
%% $ (c)\ \xymatrix{ A \ar[r] \ar@{->}[d] & A \ar[d] \\ { A\cup_{\FF}\{\text{``x''}\} } \ar[r] \ar@{-->}[ur] & \{\bullet\} }$% \
%%
%%
%% $ (d)\ \xymatrix{ X \ar[r]^{\text{id}} \ar@{->}[d] & X \ar[d]^{f} \\ { X\cup_{\mathfrak U}\{\text{``x''}\} } \ar[r] \ar@{-->}[ur] & Y }$% \
%% %$(d)\ \rrt X {\therefore(inj)} {Y} {\{x,y\}} {} {\{x=y\}}$\
%% $ (e)\ \xymatrix{ A \ar[r] \ar@{->}[d] & X \ar[d]^{g} \\ { A\cup_{\mathfrak U}\{\text{``x''}\} } \ar[r] \ar@{-->}[ur] & Y }$% \
%%
%%
%% $ (f)\ \xymatrix{ A \ar[r]^f \ar@{->}[d] & Y \ar[d]^{g} \\ { X } \ar[r] \ar@{-->}[ur] & \{\bullet\} }$% \
%% $ (g)\ \xymatrix{ A \ar[r]^f \ar@{->}[d] & Y \ar@{->}[r] & { \{\boldsymbol{B_1}\leftarrow O\rightarrow \boldsymbol{B_2}\} } \\ { X } \ar@{-->}[urr] & { } & { } }$% \
%% $ (h)\ \xymatrix{ A \ar[r] \ar@{->}[d] & { \{\boldsymbol{B_1}\leftarrow O\rightarrow \boldsymbol{B_2}\} } \ar[d] \\ { X } \ar[r] \ar@{-->}[ur] & {\{\bullet\} } }$% \
%%
%% $ (i)\ \xymatrix{ A \ar[r] \ar@{->}[d] & { \{U\} } \ar[d] \\ { X } \ar[r] \ar@{-->}[ur] & { \{ U \ra U' \} } }$% \
%% $ (j)\ \xymatrix{ A \ar[r] \ar@{->}[d] & { \{x\llrra y\} } \ar[d] \\ { X } \ar[r] \ar@{-->}[ur] & \{x=y\} }$% \
%% $ (k)\ \xymatrix{ A \ar[r] \ar@{->}[d] & { \{o \ra c \} } \ar[d] \\ { X } \ar[r] \ar@{-->}[ur] & \{o=c\} }$% \
%%
%% $ (l)\ \xymatrix{ {\{o\} } \ar[r] \ar@{->}[d] & X \ar[d] \\ { \{o \ra c \} } \ar[r] \ar@{-->}[ur] & { Y } }$% \
%%
%$(d)\ \rrt X {\therefore(inj)} {Y} {\{x,y\}} {} {\{x=y\}}$\
\end{center}
\caption{\label{fig1}\small
%Lifting properties.
These are equivalent reformulations of quasi-compactness of spaces and its generalisation to maps, that of properness of maps.
%Dots $\therefore$ indicate free variables and what property of these variables is being defined;
%, i.e.~a property of what is being defined and how is it to be labelled
%in a diagram chasing calculation, ``$\therefore(surj)$'' reads as:
%given a (valid) diagram, add label $(surj)$ to the corresponding arrow.\newline
(a) the identity map $K \xra {\id} K$
factors as $K\lra K\cup_{\FF} \{\oo\} \lra K$
%: for each $i:A\lra X$ and $j:B\lra Y$
%making the square commutative, i.e.~$f\circ j=i\circ g$, there is a diagonal arrow $\tilde j:B\lra X$ making the total diagram
%$A\xra f B\xra {\tilde j} X\xra g Y, A\xra i X, B\xra j Y$ commutative, i.e.~$f\circ \tilde j=i$ and $\tilde j\circ g=j$.
(b) this is also equivalent to $K$ being quasi-compact (we no longer require the arrow $K\lra K$ to be identity)
(c) and in fact quasi-compact spaces are orthogonal to maps associated with ultrafilters %(say better!)
(d) $X\xra f Y$ is proper, i.e. {\sf d) If $\mathfrak U$ is an ultrafilter on $X$ and if $y \in Y$ is a limit point of the ultrafilter
base $f (U)$, then there is a limit point $x$ of $\mathfrak U$ such that $f (x) = y$.} \href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki, General Topology, I\S10.2,Th.1(d)]}
(e) this is also equivalent to $X\xra f Y$ is proper, i.e. this holds for each ultrafilter $\mathfrak U$ on each space $A$ %(say better!)
(f) The mapping \ensuremath{f} has a continuous extension over \ensuremath{X}
(h) for every pair $B_1,B_2$ of disjoint closed subsets of \ensuremath{Y} the inverse images $f^{-1}(B_1)$
and $f^{-1}(B_2)$ have disjoint closures in the space $X$
(i) the image of $A$ is dense in $B$
(j) the map $A\lra B$ is injective
(k) the topology on $A$ is induced from $B$
(l) for $X$ and $Y$ finite, this means that the map $X\lra Y$ is closed, or, equivalently, proper
} \end{figure}
\subsubsection{\label{comp:m2}Compactification as factorisation system/M2-decomposition}
By a simple diagram chasing argument,\footnote{\label{foot:bous}
See Thm.~3.1 of \href{https://core.ac.uk/download/pdf/82479252.pdf}{[Bousfield, Constructions of factorization systems in categories]}
for details of such an argument and assumptions which are enough to make it work. However, note that his definitions are somewhat different from ours: unlike us, he considers the {\em unique} lifting property, cf.\S2 [ibid.].}
\footnote{
See \href{http://www.maths.uwc.ac.za/~dholgate/Papers/DBHPhd.zip}{[Holgate,PhD,2.1(Perfect Maps)]} and references therein
for examples of factorisation systems related to Stone-\' Cech decomposition and proper maps.
Note [Holgate] says ``perfect'' instead of ``proper'', as is common in topology.
}
each morphism $X\lra Y$ decomposes as either
$X\xra{(P)^{rl}}\cdot\xra{(P)^r} Y $ and $X\xra{(P)^{l}}\cdot\xra{(P)^{lr}} $
whenever $P$ is a class of morphisms and the underlying category has enough limits and colimits.
We shall now see that Stone-\v{C}ech compactification is an example of such a decomposition
when $P$ is the class of proper maps
and is thus somewhat %can be constructed by a Quillen small object argument
analogous to the $(cw)(f)$- and $(c)(wf)$-decomposition
required by Axiom M2 of Quillen model categories.
Almost this observation is mentioned explicitly in \href{https://core.ac.uk/download/pdf/82479252.pdf}{[Bousfield, Constructions of factorization systems in categories]}\footnote{
In our notation $\mathcal M (\mathbf E_1)$ is almost $(\mathbf E_1)^r$ but not quite:
$\mathcal M (\mathbf E_1)$ is the right orthogonal ($\rtt$-negation) with respect to the {\em unique} lifting property; [7] is [S. MacLane, Categories for the Working Mathematician (Springer-Verlag, New York, 1971)].
}:
\newline\noindent\includegraphics[width=\linewidth]{bousfield-fact-systems-51-ex.png}
%%%- \begin{quote}
%%%- {\bf 5.1 Example.} In the category $\bf Top$ of topological spaces, let $\mathbf E_1$ be the class of all
%%%- maps $X\lra Y$ inducing a bijection ${\bf Top}(Y, I) \approx {\bf Top}(X, I)$ where $I$ is the closed unit
%%%- interval. Then $({\mathbf E}_1, \mathcal M (\mathbf E_1))$ is a factorization system in $\bf Top$ by 3.4. One can show
%%%- that the $({\mathbf E}_1, \mathcal M (\mathbf E_1))$-localization (2.5) on $\bf Top$ is just the Stone-Cech compactification (cf. [7, p. 127]).
%%%- \end{quote}
%%%-
\vskip-7pt
\noindent We shall find this observation by transcribing [Engelking, Theorem 3.6.1, p.173] by means of diagram chasing.
In fact, corollaries [Engelking, 3.6.2-3.6.9] could also be seen in a diagram chasing way; we only reformulate Corollary~3.6.3.
\begin{figure}%\newline
\noindent\includegraphics[width=\linewidth]{Engelking-36-stone.png}
\newline\noindent\includegraphics[width=\linewidth]{Engelking-361-compactification.png}
\newline\noindent\includegraphics[width=\linewidth]{Engelking-361-colls.png}
\newline\noindent\includegraphics[width=\linewidth]{Engelking-Coll-363.png}
\vskip5pt\small
\begin{center}
$ (a)\ \xymatrix{ X \ar[r]|{\text{f}} \ar@{->}[d] & Z \ar[d]|{(P)} \\ { \beta X } \ar[r]|{(P)} \ar@{-->}[ur]& \{\bullet\} }$% \
\ \ \ \ $ (b)\ \xymatrix{ X \ar[r]|{\forall} \ar@{->}[d] & {\beta X} \ar[d]|{(P)} \\ { \alpha X } \ar[r]|{(P)} \ar@{-->}[ur]& \{\bullet\} } $% \
\ \ \ \ $ (b')\ \xymatrix{ X \ar[r]|{(P)^l} \ar@{->}[d]|{(P)^l} & {\cdot} \ar[d]|{(P)^{lr}} \\ {\cdot} \ar[r]|{(P)^{lr}} \ar@{-->}[ur]|{(iso)}& \{\bullet\} } $% \
$ (c)\ \ \ X\xra{(P^l)} \beta X \xra {(P)} \{\bullet\}$
\ \ \ \ $ (d)\ \ \ \ X\xdra{(P^l)} Y \xdra {(P)^{lr}} Z$\\
$ (e) \ \ \ \ X\xra{(P^l)} \beta X \rtt [0,1]\lra\{\bullet\}$\\
$(f)\ \ \ \ \ \ X\lra \alpha X \xra {(P)} \{\bullet\}$ and $ X\xra{(P^l)} \alpha X \rtt [0,1]\lra\{\bullet\}$ implies $\alpha X = \beta X$
%\xymatrix{ X \ar[r]|{(P^l)} & \beta X \ar[r]|{(P)} & {\{\bullet\}}}$
\end{center}
\caption{\label{fig2}\small
A diagram chasing reformulation of [Engelking, Theorem 3.6.1, p.173].
(a) {\sf Every continuous mapping $f: X \lra Z$ of a Tychonoff space $X$ to a compact
space Z is extendable to a continuous mapping $F: \beta X \lra Z$.}
(b) {\sf If every continuous mapping of a Tychonoff space X to a compact space is
continuously extendable over a compactification $\alpha X$ of $X$, then $\alpha X$
is equivalent to the Cech-Stone
compactification of $X$.} This is reformulated as follows: if diagram $(b)$ holds,
then the diagonal map $\alpha X\lra \beta X$ can be chosen to be an isomorphism.
(b$'$) this is an analogue of (b) formulated in terms of category theory as uniqueness of $\cdot\xra{(P)^l}\cdot\xra{(P)^{lr}}\cdot$ decomposition;
the diagonal arrow exists because $(P)^l\rtt (P)^{lr}$
and thus we require it to be an isomorphism.
(c) Both diagrams above can summarized as: there exists a unique decomposition of this form.
(d) Further, this is implied by an analogue of Axiom M2 $(cw)(f)$- and $(c)$$(wf)$-decomposition
of model categories: each morphism $X\lra Z$ decomposes as
$ X\xdra{(P^l)} Y \xdra {(P)^{lr}} Z$
(e) %3.6.3. COROLLARY. %
{\sf Every continuous function $f: X \lra X$ from a Tychonoff space $X$ to the
closed interval $I$ is extendable to a continuous function $F: \beta X\lra I$.}
(f) {\sf If every continuous function from a Tychonoff space $X$ to the closed interval $I$ is
continuously extendable over a compactification $\alpha X$ of $X$, then $\alpha X$ is equivalent to the \v Cech-Stone
compactification of $X$.} Note the conclusion $\alpha X=\beta X$ is stated somewhat imprecisely; we rather need
to say that morphisms $X\lra \alpha X$ and $X\lra \beta X$ are the same.
}\end{figure}
Let us transcribe this by means of the notation of arrows. We will deliberately ignore the separability assumptions that
$X$ is Tychonoff and $\beta X$ and $Z$ are assumed to be Hausdorff.
Let $P$ be the class of proper maps.
Figure 2ab represent the statement of the theorem.
Figure 2a suggests that the compactification map $X\lra \beta X$ is in the class $(P)^l$;
Figure 2b suggests that there is a unique decomposition
$X\xra{(P^l)} X' \xra {(P)^{lr}} \{\bullet\}$.
And indeed, this is implied by a simple diagram chasing argument.
Uniqueness follows from orthogonality of $(P)^l$ and $(P)^r$.
The decomposition is constructed by an argument which looks roughly as follows:\footnote{For details see footnote ${}^{(\ref{foot:bous})}$.}
%Thm.~3.1 of \href{https://core.ac.uk/download/pdf/82479252.pdf}{[Bousfield, Constructions of factorization systems in categories]}...
consider all the decompositions of form $X\xra{(P^l)} X'\lra Y$ and
take the pushout $X\xra{(P)^l} X_l$ of all the maps $X\xra{(P^l)} X'$ appearing in the decompositions of this form.
The map belongs to $(P)^l$ because left orthogonals are closed under pushouts,
By the universality property of pushouts you obtain a decomposition $X\xra{(P)^l} X_l\lra Y$
and a diagram chasing argument based on the definition of pushout and orthogonality properties of $(P)^l$ and $(P)^{lr}$
shows the map $X_l\lra Y$ is right orthogonal to $(P)^{l}$, i.e.~belongs to $(P)^{lr}$ as required.
An argument of this kind is known as Quillen small object argument and originally was
used to prove Axiom M2 $(cw)(f)$- and $(c)(wf)$-decomposition of model categories.
The argument shows that under suitable assumptions that a category has enough
limits and colimits, any morphism $X\lra Z$ decomposes as
$X \xra{(P^l)} Y \xra {(P)^{lr}} Z$, for any class $(P)$ of morphisms.
Here we take $(P)$ to be the class of proper morphisms.
%$Z\xra{(P)}\{\bullet\}$ implies $X\xra f Z $ factors as $X\xra ? \beta X \xra F Z $
%$X\xra g Y \xra{(P)}\{\bullet\}$ implies $X\xra g Y$ factors as $X\lra \alpha X \lra Y$, then
%maps $X\lra \alpha X$ and $X\lra \beta X$ are equal.
%%%---
%%%--- A simple diagram chasing argument shows that
%%%--- each morphism $X\lra Y$ decomposes as $X\xra{(P)^l} X' \xra {(P)^{lr}} Y$
%%%--- if the category has enough limits and colimits. This argument is known as
%%%--- Quillen small object argument (for the purposes of the argument)..
%%%---
\vskip15pt\noindent
We end our discussion of compactness with the following rather vague considerations;
we hope they might suggest the reader something about the arrow notation (calculus)
appropriate for topology. We admit that what we say below is very vague.
\subsubsection{\label{comp:AEEA}Compactness as being uniform. $\forall\exists\implies\exists\forall$}
Often an application of compactness is as follows.
We know that certain choices can be made for each value of parameters;
if we also know that the parameters vary over a compact domain, then
we may assume that these choices are uniform,
i.e.~that they do not depend on the value of the parameters.
Put another way, compactness allows to change the order of quantifiers
$\forall\exists\implies\exists\forall$ in certain formulas. See Appendix~\ref{app:AEEA}
for a list of examples.\footnote{For a discussion see Remark~8 of
\href{http://mishap.sdf.org/mints/expressive-power-of-the-lifting-property.pdf}{[Gavrilovich, Lifting Property]}
}
The subsection of \href{https://stacks.math.columbia.edu/tag/005M}{[Stacks Project, I.5\S15, tag 005M]} dealing with
the Bourbaki characterisation of proper maps
starts with a lemma of this kind:
%Part 1: Preliminaries Chapter 5: Topology Section 5.17: Characterizing proper maps (cite)
%%%--- \section{Characterizing proper maps}
%%%--- \label{section-proper}
%%%--- \noindent
%%%--- We include a section discussing the notion of a proper map in usual
%%%--- topology. It turns out that in topology, the notion of being proper
%%%--- is the same as the notion of being universally closed, in the sense
%%%--- that any base change is a closed morphism (not just taking products
%%%--- with spaces). The reason for doing this is that in algebraic geometry
%%%--- we use this notion of universal closedness as the basis for our
%%%--- definition of properness.
%%%--- \begin{lemma}[Tube lemma]
%%%--- \label{lemma-tube}
%%%---
\begin{lemma}[Tube lemma]
\label{lemma-tube}
Let $X$ and $Y$ be topological spaces.
Let $A \subset X$ and $B \subset Y$ be quasi-compact subsets.
Let $A \times B \subset W \subset X \times Y$ with $W$
open in $X \times Y$. Then there exists opens $A \subset U \subset X$
and $B \subset V \subset Y$ such that $U \times V \subset W$.
\end{lemma}
In a somewhat more old-fashioned way,
this lemma can be reformulated as follows:
\begin{lemma}[Tube lemma]
\label{lemma-tube}
Let $X$ and $Y$ be topological spaces.
Let $A \subset X$ and $B \subset Y$ be quasi-compact subsets.
Let $A \times B \subset W \subset X \times Y$. % with $W$ open in $X \times Y$.
%Then there exists opens $A \subset U \subset X$
%and $B \subset V \subset Y$ such that $U \times V \subset W$.
If for each pair of points $a\in A$ and $b\in B$
we can pick neighbourhoods $U=U(a,b)\ni a$ and $V=V(a,b)\ni b$ such that
$(a,b)\in U\times V\subset W$, then we can do so uniformly in $a\in A$ and $b\in B$,
i.e.~such that $U=U(a,b)$ and $V=V(a,b)$ do not depend on $a$ and $b$.
As a formula, this could be expressed as change of order of quantifiers:
$$\frac{
\forall a\in A \forall b \in B\, \exists U\subset X \, \exists V \subset Y \,(U\times V\subset W\text{ and }a\in U\text{ is open and }b\in V\text{ is open})
}{
\exists U\subset X\, \exists V\subset Y\, \forall a\in A \forall b \in B \,(U\times V\subset W\text{ and }a\in U\text{ is open and }b\in V\text{ is open})
}$$
\end{lemma}
The following example of change or order of quantifiers is simpler but perhaps more telling.
%\begin{lemma}
For a connected topological space $X$, the following are equivalent:
\bi
\item Each real-valued function on $X$ is bounded
\item $\forall x \in K \exists M ( f(x) < M ) \implies \exists M \forall x \in K ( f(x) < M ) $
\item $\emptyset \longrightarrow K \rtt \sqcup_{ n\in\NN} (-n,n) \longrightarrow \RR$\\
here $\cup_ n (-n,n) \longrightarrow \RR$ denotes the map to the real line
from the disjoint union of intervals $(-n,n)$ which cover it.
Note this is a standard example of an open covering of $\RR$ which
shows it is not compact.
\ei
The following is even more vague.
\subsubsection{ %tex ###
``An open covering has a finite subcovering''
%"an open neighbourhood U(x) of a point x as a function of the point"
} %tex
Mathematically, this reformulation is based on the following observation:
\def\ooo{\infty}
\begin{quote}
a space $K$ is compact iff for each open covering $U$ of $K$,
the subset $K$ is closed in $K\cup\{\ooo\}$ in the topology generated
elements of $U$ as {\em closed} subsets.
\end{quote}
%%%This uses that the topology is generated by {\em finite} unions
%%%of basic closed subsets.
%%%
%%%alternative:-
This lets us express being {\em finite} with the help of
the notion of the topology generated by a family of sets.
%%%%
%%%%(fixme: skip the explanation below??)
%%%%Indeed, if K is compact, then K is the union of elements
%%%%of a finite subcovering, hence is closed in this topology;
%%%%conversely, if K is a closed subset of K\cup\{oo\}, then
%%%%K can be represented as a finite union of intersections
%%%%of sets in U, hence a finite union of sets in U,
%%%%which mean these sets form a finite subcovering.
%%%%
%%
[Hausdorff, Set theory] denotes by $U(x)$ a neighbourhood of a point $x$,
which suggests viewing $U(x)$ as a (possibly multivalued) function of a point $x$ ;
We'd like to develop ``arrow'' notation where this would be expressed as
%%%%An (open) covering >implicitly defines|maybe thought?
%%%%is often thought of as a (multivalued) function
%%%%assigning an (open) subset to each point x in K.
%%%%
%%%%In the notation we would like to develop,
%%%%we would the following to denote an open covering U :
%%%%
%%
$$\{x\}\longrightarrow K\xra{(U(x))} \{x\rightarrow y\}\ \ \ \ \ \ \ \ \ (*)$$
here it is implicit that $x$ maps to $x$ by the composition of the two arrows;
``$(x)$'' in ``$U(x)$'' signifies that $U(x)$ depends on $x$.
Changing a single symbol ``$\rightarrow$'' into ``$\leftarrow$'' leads us to consider
elements of $U$ as closed subsets of $K$:
%
$$\{x\}\longrightarrow K\xra{(U(x))} \{x\leftarrow y\}\ \ \ \ (**)$$
We'd like to assume (or require) that $(**)$ inherits some properties of $(*)$,
in the arrow calculus we'd like to define;
this would be what corresponds to considering {\em the topology generated by}.
%%%--- Manipulating a definable family of arrows from an $X$,
%%%--- by its nature, is similar to working
%%%--- with the topology on $X$ satisfying properties
%%%--- which reflect what we use about the family.
%%%---
%%%Manipulating arrows (**) is to be equivalent to
%%%working with the topology generated by the sets in U
%%%as closed sets.
%%%--- Of course, this change of a symbol would only make sense
%%%--- within a context of a formal calculus %%|derivation system|formal notation?
%%%--- which we do not have yet.
%%%--- In our calculus, the arrows $(**)$
%%%--- should inherit some properties from $(*)$, e.g. a family of arrows $(**)$
%%%--- commutes iff the corresponding family of arrows $(*)$ commutes.
%%%--- ..A collection of arrows of form $(**)$ (or $(*)$) should
%%%--- define a topology generated by sets in $U$....
%%%--- ....
%%%---
\subsubsection{Summary.} These three examples suggest
that orthogonality, or $\rtt$-negation,
%uniqious and implicit in many standard definitions. See Appendix~\ref{app:rtt-examples} and Appendix~\ref{app:rtt-top}
has a surprising generative power
%of the lifting property (orthogonality of morphisms in a category)
as a means of
defining natural elementary mathematical concepts.
In Appendix~\ref{app:rtt-examples} and Appendix~\ref{app:rtt-top} we give a number of examples in
various categories, in particular showing that many standard elementary notions
of abstract topology can be defined by applying the lifting property to simple
morphisms of finite topological spaces.
Examples in topology include the notions of: compact, discrete, connected, and
totally disconnected spaces, dense image, induced topology, and separation axioms.
Examples in algebra include: finite groups being
nilpotent, solvable, torsion-free, $p$-groups, and prime-to-$p$ groups;
injective and projective
modules; injective, surjective,
and split homomorphisms.
\subsection{Hausdorff axioms of topology as diagram chasing computations with finite categories
\label{ax:chasing}}
We shall now reformulate the axioms of a topology in a form
almost ready to be implemented in a theorem prover based on diagram chasing.
%Above we reformulated a number of elementary notions in topology in terms of preorders.
%Now we observe that the original axioms of topology as formulated by Hausdorff
%can also be viewed as rules for manipulating
%finite preorders.
Early works talk of topology in terms of {\em neighbourhood} systems $U_x$
where $U_x$ varies though
{\em neighbourhoods of points} of a topological space. This is how
the notion of topology was defined by Hausdorff; in words of [Bourbaki],
``We shall say that a set $E$ carries a topological structure whenever we have
associated with each element of $E$, by some means or other, a family
of subsets of $E$ which are called neighbourhoods of this element - provided
of course that these neighbourhoods satisfy certain conditions (the axioms
of topological structures).''
Whenever we are speaking of a neighbourhood $U_x$ of a point $x\in E$, we are speaking of % determines
two functions
$$
\{x\}\longrightarrow X\xrightarrow{\,\,\, U\,\,\,}\{x \ra x'\}$$
We would like to be able to say that a set $E$ carries a topological structure whenever we have
associated with each element $x$ of $E$, by some means or other,
a family
of %subsets of $E$ which are called neighbourhoods of this element -
arrows, or functions,
$\{x\}\lra E\lra \{x\ra x'\}$
provided
of course that these arrows %neighbourhoods
satisfy certain conditions (corresponding to the axioms
of topological structures).
%This allows to reformulate the axioms of topology as rules of diagram chasing with
%finite categories.
%{\em a neighbourhood system $U_x$, $x\in X$} would correspond to a system of arrows
%$$
%\{x\}\longrightarrow X\xrightarrow{\,\,\, U\,\,\,}\{x{\small\searrow}x'\}$$
%and Hausdorff's axioms (A),(B),(C) (see Appendix~B) would correspond to
%diagram chasing rules.
This simple observation allows us to show that
the axioms of topology formulated in the more modern
language of open subsets %The Hausdorff axioms of a topological space
can be seen as diagram chasing rules for manipulating diagrams
involving notation such as
$$ \{x\}\longrightarrow X\ \ \ \ X\longrightarrow \{x{\small\searrow}y\}\ \ \ \ X\longrightarrow \{x\leftrightarrow y\} $$
in the following straightforward way. %; cf. [Gavrilovich, Elementary Topology,\S.2.1] for more details.
\subsubsection{Axioms of open sets as diagram chasing rules.}
As is standard in category theory, identify a point $x$ of a topological space $X$
with the arrow $\{x\}\longrightarrow X$, a subset $Z$ of $X$ with the arrow $X\longrightarrow \{z\leftrightarrow z'\}$,
and an open subset $U$ of $X$ with the arrow $X\longrightarrow \{u{\small\searrow}u'\}$.
With these identifications, the Hausdorff axioms of a topological space become
rules for manipulating such arrows, as follows.
{\em Both the empty set and the whole of \ensuremath{X} are open} says that the compositions
$$ X\longrightarrow \{c\}\longrightarrow \{o{\small\searrow}c\}\text{ and }X\longrightarrow \{o\}\longrightarrow \{o{\small\searrow}c\} $$
behave as expected (the preimage of \{o\} is empty under the first map,
and is the whole of \ensuremath{X} under the second map).
{\em The intersection of two open subsets is open} means the arrow
$$ X\longrightarrow \{o{\small\searrow}c\}\times\{o'{\small\searrow}c'\} $$
behaves as expected (the ``two open subsets'' are the preimages of points $o\in\{o{\small\searrow}c\}$ and $o'\in\{o'{\small\searrow}c'\}$;
``the intersection'' is the preimage of $(o,o')$ in
$\{o{\small\searrow}c\}\times\{o'{\small\searrow}c'\}$ ).
\vskip4pt
{\em The preimage of an open set is open} says the composition $$
X\longrightarrow Y\longrightarrow \{u{\small\searrow}u'\}\longrightarrow \{u\leftrightarrow u'\} $$
is well-defined.
We need the following terminology to formulate the next diagram chaning reformulation.
We say that {\em a diagram commutes from vertex $A$ to vertex $B$} iff the composition of morphisms along any two paths from $A$ to $B$ is the same.
We say {\em a diagram commutes at (to) a vertex $A$} iff it commutes from $A$ to any vertex $B$ (from any vertex $A$ to $B$, resp.).
Finally, let us write a diagram chasing rule which corresponds to the fact that
in topology we consider subsets which of elements and that functions are
defined element-wise. It allows to reduce diagram chasing to finite objects.
\bi\item
for each arrows $A\xra f B$, $A\xra g B$ it holds\\
% $ \xymatrix{ {\ \ \ \ \ \ \ \ } & X \ar[d] \\
%{ A } \ar[r] \ar[ur] & { Y }}$
$ \xymatrix{A \ar@/^/[r]|f\ar@/_/[r]|g & B}$
\ \ \ \ \ \ \ \ \begin{minipage}[c]{0.3\textwidth}
iff for each $\{u\}\lra A$, \\
the diagram commutes at vertex $\{u\}$:
\end{minipage}\ \ \ \ \
$ \xymatrix{\{u\}\ar[r] & A \ar@/^/[r]|f\ar@/_/[r]|g & B} $
\item
for each arrows
$A\xra f B$, $A\xra g B$ it holds\\
$ \xymatrix{A \ar@/^/[r]|f\ar@/_/[r]|g & B}$
\ \ \ \ \ \ \ \ \begin{minipage}[c]{0.3\textwidth}
iff\\ for each $B\lra \{x\llrra y\}$, \\
the diagram commutes to vertex $\{x\llrra y\}$:
\end{minipage}\ \ \ \ \
$ \xymatrix{A \ar@/^/[r]|f\ar@/_/[r]|g & B \ar[r] & \{x\llrra y\} } $
\item
for each arrows $A\xra f B$, $A\xra g B$ it holds\\
% $ \xymatrix{ {\ \ \ \ \ \ \ \ } & X \ar[d] \\
%{ A } \ar[r] \ar[ur] & { Y }}$
$ \xymatrix{A \ar@/^/[r]|f\ar@/_/[r]|g & B}$
\ \ \ \ \ \ \ \ \begin{minipage}[c]{0.3\textwidth}
iff for each $\{u\}\lra A$ and $B\lra \{x\llrra y\}$, \\
the diagram commutes from $\{u\}$ to $ \{x\llrra y\}$ :
\end{minipage}\ \ \ \ \
$ \xymatrix{\{u\}\ar[r] & A \ar@/^/[r]|f\ar@/_/[r]|g & B \ar[r] &\{x\llrra y\} }$
\ei
%%%- Finally, {\em a subset $U$ of $X$ is open iff each point $u$ of $U$ has an open
%%%- neighbourhood inside of $U$}
%%%- corresponds to the following diagram chasing rule:
%%%-
\subsubsection{\label{ax:union:AEEA}An arbitrary union of open subsets is open as $\forall\exists\implies\exists\forall$% quantifier exchance.
.}
Finally, the axiom that an arbitrary union of open subsets is necessarily open
can be reformulated in the following ways:
\bi
\item
{\em A subset $U$ of $X$ is open iff each point $u$ of $U$ has an open
neighbourhood inside of $U$}.
%- \item
%- {\em If for each point $u$ of a subset $U$
%- we can pick an open neighbourhood $u \in U_u \subseteq U$ within $U$,
%- then we can do so in such a way that $U_u$ does not depend on $u$}
%- (and therefore $U_u=U$ for each $u\in U$).
\item
{\em If for each point $x$ of a subset $U$
we can pick an open neighbourhood $x \in U_x \subseteq U$ within $U$,
then we can do so in such a way that $U_x$ does not depend on $x$}
(and therefore $U_x=U$ for each $x\in U$).
\item
As an $\forall\exists\implies\exists\forall$ implication (cf.~\S\ref{comp:AEEA},\S\ref{app:AEEA}),
$$\frac{
\forall x \in U\, \exists V \, ( x \in V \text{ and } V\subseteq U \text{ and } V\text{ is open} \,)}{
\exists V\,\forall x \in U \, ( x \in V \text{ and } V\subseteq U \text{ and } V\text{ is open} \,)}
$$
%\end{enonce}
\ei
Let us give several reformulations in terms of diagram chasing:
\vskip4pt
%+for each arrow h:X\longrightarrow \{U\leftrightarrow U'\},
%+
%+ \{U{\small\searrow}U'\}
%+ \cap |
%+(exist) / |
%+ / |
%+ / |
%+ / v
%+ \ensuremath{X} --h\longrightarrow \{U\leftrightarrow U'\}
%+
%+iff
%+
%+
%+\{u\}---\longrightarrow \{u{\small\searrow}U\leftrightarrow U'\}
%+ | \cap |
%+ |(exist)/ |
%+ | / |
%+ | / |
%+ \ensuremath{v} / v
%+ \ensuremath{X} --h\longrightarrow \{u=U\leftrightarrow U'\}
%+
%+here the arrows $\{U\}\longrightarrow \{U{\small\searrow}U'\}$
%+and $\{U{\small\searrow}U'\}\longrightarrow \{U\leftrightarrow U'\}$ denote the obvious maps.
%\input diagram_open_subsets_3arrow.tex
for each arrow $X\xra{\xi_U} \{U\leftrightarrow \bar U \}$ it holds\\
$ \xymatrix{ {\ \ \ \ \ \ \ \ } & \{U\ra\bar U\} \ar[d] \\
{\ \ \ \ X\ } \ar[r]|-{\xi_U} \ar@{-->}[ur] & { \{U\leftrightarrow \bar U \} }}$
\ \ \ \ \ \ iff for each $\{u\}\lra X$ \ \ \ \
$ \xymatrix{ {\{u\}} \ar[r] \ar[d] & {\{u \ra U \leftrightarrow \bar U\}} \ar[d]^{\ \ \ \ \ \ \ \ \ \ \ (\star)} \\
{\ \ \ X\ \ \ } \ar[r]|-{\xi_U} \ar@{-->}[ur] & { \{u\!=\!U\!\leftrightarrow\! \bar U \} }}$
\vskip6pt
The following reformulation uses that sets consist of points:
for each arrow $X\xra{\xi_U} \{U\leftrightarrow \bar U \}$ the following are equivalent:\\
\bi
\item
\begin{minipage}[t]{0.5\textwidth}
there is an arrow $X\lra \{U\lra \bar U\}$\\
for each $\{u\}\lra X$\\
the diagram commutes at vertex $\{u\}$
\end{minipage}
$ \xymatrix{ \{u\} \ar[d] & \{U\ra\bar U\} \ar[d] \\
{X} \ar[r]|-{\xi_U} \ar@{-->}[ur] & { \{U\leftrightarrow \bar U \} }}$
\item
\begin{minipage}[t]{0.5\textwidth}\small
for each $\{u\}\lra X$\\
%for each $\{u\}\lra \{u\ra U\llrra \bar U\}$\\
making the square commute\\
there is an arrow $X\lra \{u\ra U\llrra \bar U\}$\\
making the diagram commute
\end{minipage}
$ \xymatrix{ {\{u\}} \ar[r]^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!u\mapsto u} \ar[d] & {\{u \ra U \leftrightarrow \bar U\}} \ar[d] \\
{ X } \ar[r]|-{\xi_U} \ar@{-->}[ur] & { \{u\!=\!U\!\leftrightarrow\! \bar U \} }}$
\item
\begin{minipage}[t]{0.5\textwidth}\small
for each $\{u\}\lra X$\\
for each $\{u\}\lra \{u\ra U\llrra \bar U\}$\\
making the square commute\\
there is an arrow $X\lra \{u\ra U\llrra \bar U\}$\\
making the diagram commute
\end{minipage}
$ \xymatrix{ {\{u\}} \ar[r] \ar[d] & {\{u \ra U \leftrightarrow \bar U\}} \ar[d] \\
{ X } \ar[r]|-{\xi_U} \ar@{-->}[ur] & { \{u\!=\!U\!\leftrightarrow\! \bar U \} }}$
\ei
\vskip6pt
\begin{enonce}{Question}
It is tempting to rewrite this as a following sequence of diagram chasing rules,
which seem to be closer to intuitive considerations:
\bi
\item pick a new arrow $\{u\}\lra X$
\item construct an arrow $X\lra \{u\ra U\llrra \bar U\}$ in some way
such that the diagram $(\star)$ commutes
\item remove the dependency of $X\lra \{u\ra U\llrra \bar U\}$ on $\{u\}\lra X$, i.e.~label the arrow $X\lra \{u\ra U\llrra \bar U\}$ as {\em not dependent}
on $\{u\}\lra X$
\ei
Define a formal syntax and a proof system which captures this kind of derivationns.
\end{enonce}
We cannot stress enough the speculation below.
\begin{enonce}{Remark}\label{top:AEEA} We find it extremely intriguing that an axiom of topology
%that {\em a union of open sets is open}
admits an $\forall\exists\implies\exists\forall$ reformulation. We view it as a sign that
topology is really about, so to say, permuting quantifiers, or, in other words,
expressly manipulating what variable/term/construction depends on what; so to say,
topology reasons about dependency rather than continuity. A
technical way to start thinking about this point of view is provided by the
$\forall\exists\implies\exists\forall$ reformulations in \S\ref{ax:union:AEEA}
and $\forall\exists\implies\exists\forall$ reformulations of compactness in
~\S\ref{comp:AEEA} and \S\ref{app:AEEA}. See also a discussion in \S\ref{q:AEEA}.
\end{enonce}
\vskip8pt
%
%\begin{itemize}
%\item[]
%
%\item[($3_\rightarrow$)] %a set $U$ is open iff for every point $u\in U$ there is an open subset $U'$ such that $u\in U'\subset U$.
%for each arrow $X\xra[\xi_U]\\{\\} \{U\llrra \bar \ensuremath{U} \}$ it holds\\
% $ \xymatrix{ \{ \} & \{U\ra\bar U\} \ar[d] \\
%\{X\} \ar[r]_{\!\!\!\!\!\!\!\!\!\xi_U} \ar@{\longrightarrow }[ur] & { \{U\llrra \bar \ensuremath{U} \} }}$
%\ \ \ \ iff for each $\{u\}\lra X$, \ \ \ \ \ \ \
% $ \xymatrix{ {\{u\}} \ar[r] \ar[d] & \{u \ra \ensuremath{U} \llrra \bar U\} \ar[d] \\
%\{X\} \ar[r]_{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\xi_U} \ar@{\longrightarrow }[ur] & { \{u=U\llrra \bar \ensuremath{U} \} }}$
%
%\item
%
%a set $U$ is open iff for every point $u\in U$ there
%is an open subset $U'$ such that $u\in U'\subset U$.
%for each arrow $X\xra[\xi_U]\\{\\} \{U\longleftrightarrow \bar \ensuremath{U} \}$ it holds\\
% $ \xymatrix{ \{ \} & \{U\ra\bar U\} \ar[d] \\
%\{X\} \ar[r]_{\!\!\!\!\!\!\!\!\!\xi_U} \ar@{ \longrightarrow }[ur] & { \{U\longleftrightarrow \bar \ensuremath{U} \} }}$
%\ \ \ \ iff for each $\{u\}\lra X$, \ \ \ \ \ \ \
% $ \xymatrix{ {\{u\}} \ar[r] \ar[d] & \{u \ra \ensuremath{U} \longleftrightarrow \bar U\} \ar[d] \\
%\{X\} \ar[r]_{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\xi_U} \ar@{ \longrightarrow }[ur] & {
%\{u=U\longleftrightarrow \bar \ensuremath{U} \} }}$
%\end{itemize}
%
%\begin{enonce}{Remark}
%This observation suggests that some arguments in elementary topology may be
%understood entirely in terms of diagram chasing, see [Gavrilovich, Elementary Topology] for some
%examples.
We hope that this reinterpretation may help clarify the nature of the axioms of a
topological space, in particular it offers a constructive approach and a diagram chasing
formalisation of certain elementary arguments, may clarify
to what extent set-theoretic language is necessary, and perhaps help to suggest
an approach to ''tame topology'' of Grothendieck,
%, Does this lead to tame topology of Grothendieck,
i.e.~a foundation of topology "without false
problems" and "wild phenomena" "at the very beginning".
\section{Topological spaces as simplicial filters.}
We shall now introduce terminology which we feel allows us to more directly give
precise meaning to phrases such as ``{\em such and such a property holds for all points sufficiently near $a$}''.
We will do so by ``reading it off'' the informal considerations of [Bourbaki, Introduction]
of the intuitive notions of limit, continuity and neighbourhood.
This section may be read independently of the rest of the paper. Unlike the previous section,
we do not introduce a concise formal syntax to describe the categorical structures that arise.
\subsection{Reading the definition of topology.}
Now we pretend to directly transcribe the following explanations of Bourbaki of the intuition of topology
and analysis \href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki, Introduction, p.13]}\newline\noindent
\includegraphics[width=\linewidth]{Bourbaki-top.png}
\includegraphics[width=\linewidth]{Bourbaki-top-uniform-intuition.png}
%%+
%%+ \begin{quote}
%%+ `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.'
%%+ ....
%%+ `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}.'
%%+ \end{quote}
%%+
%So let us treat ``{\sf whenever $x$ is sufficiently
%near $a$, $x$ has the property $P\{x\}$}'' as a {\em definition}:
Call %call
a subset $P\subseteq X\times X$ {\em topoic}
iff $(a,x)\in P$ {\sf whenever $x$ is sufficiently
near $a$}, i.e.~for each $a\in X$ there is a neighbourhood $a\in U_a\subseteq X$
of $a$ such that $\{a\}\times U_a\subseteq P$.
%using set theoretic language,
This terminology {\sf enables us to give precise meaning to the
phrase ``}for each point $a$ {\sf such and such a property [ $P(a,x)$ ] holds for all points {\em sufficiently near} $a$'':
by definition this means that the set of }pairs of {\sf points which have this property is}
topoic {\sf for the topological structure in question.}
Call a subset $P\subset X\times X\times X$ {\em topoic}
iff we can ensure $(a,x,y)\in P$ by first picking $x$ sufficiently near to $a$, and then picking $y$ sufficiently near $x$,
i.e.~for each $a\in X$ there is a neighbourhood $a\in U_a\subseteq X$ of $a$ such that
for each $x\in U_a$ there is a neighbourhood $x\in U_{a,x}\subseteq X$ such that
$(a,x,y)\in P$ for each $y\in U_{a,x}$. Similarly we define topoic subsets of $X^n$ for $n>2$, see \S\ref{def:topoic-top}
for details.
Topoic subsets of $X^n$ form a filter where by a {\em filter} we mean a topological space such that a subset containing
a non-empty open set is necessarily open. Axioms of topology say that
the Cartesian powers of $X$ with these filters
form a 2-dimensional
simplicial object $\ttt(X)$ in the category of filters.\footnote
{We find it convenient to allow filters where the empty set is big,
i.e.~we allow the filter of all subsets of a set.
The category $\Filt$ of filters can be thought in three equivalent ways:
(i) it is a full subcategory of the category of topological spaces whose
objects are spaces such that a subset containing a non-empty open subset is
necessarily open
(ii) its objects are pointed topological spaces with morphisms being maps
continuous at the point
(iii) its objects are sets equipped with a finitely additive
measure taking only two values $0$ and $1$ and such that a subset of a measure
0 set has necessarily measure 0; morphisms are measurable maps preserving the measure
(iv)
its objects are sets equipped with a collection of subsets called {\em big}
such that the intersection of two big subsets is big and a subset containing a
big subset is necessarily big as well; morphisms are maps such that the
preimage of a big subset is necessarily big.
(ii) and (iii) suggest that one may also consider the category $\FFilt$ of filters
localised as follows: two maps $f,g:X\lra Y$ are considered equal as morphisms iff they are
equal locally, resp.~almost everywhere, i.e.~the subset $\{x: f(x)=g(x)\}$ is big in $X$.}
Moreover
$$\ttt:\Topp \lra Func( \Ord{\omega}^{op}\,,\,\Filt)$$
is a fully faithful embedding of the category of topological spaces
into the category of simplicial objects (in fact, of dimension 2) in the category of filters.
Here
$\Filt\subset\Topp$ is the
%category of filters as a
full subcategory of the category of topological spaces formed by filters (for us a filter is a topological space such that
a set containing a non-empty open subset is
necessarily open, hence by a morphism of filters we mean a map such that the preimage of a big set is big). or, equivalently,
it is the category of pointed topological spaces and morphisms are maps continuous at the point and preserving the point.
$\Ord{\omega}$ denotes the category of categories corresponding to finite linear orders
$$\bullet_1\lra .. \lra \bullet_n,\ 0< n <\omega.$$
In particular,
the axiom
{\sf a neighbourhood of a point $x$ is also a neighbourhood of all points sufficiently near to $x$},
%or, equivalently, an arbitrary union of open subsets is open.
says that $pr_{1,3}:X\times X\times X\lra X\times X, \ (x_1,x_2,x_3)\mapsto (x_1,x_3)$
is continuous wrt the filters defined above, or, equivalently, wrt the filter on $X\times X\times X$ defined as
pullback along projections $X\times X \xleftarrow {pr_{1,2}}X\times X\times X\xra{pr_{2,3}}X\times X$.
Continuity of a function $f:X\lra X'$ ``{\sf given} any point $x_0\in X$ and {\sf any neighbourhood $V'$
of $f (x_0)$ in $X'$, there is a neighbourhood $V$ of $x_0$ in $X$ such that the relation
$x \in V$ implies $f (x) \in V'$}\,'' is expressed by saying that the preimage $f^{-1}(P)\subset X\times X$
of any topoic subset $P\subset X'\times X'$ is topoic, or, by saying
that
the obvious natural transformation $\ttt(X)\xra{\ttt(f)}\ttt(Y)$ is well-defined.
%Definition I
%of continuity of a function $f:X\lra X'$ {\sf may be restated in the following more intuitive form: to say
%$f$ is continuous at the point $x_0$
%means that
% $f(x)$ is as near as we
%please to $f (x_0)$ whenever $x$ is sufficiently near $x_0$}.
%%+ a neighbourhood of $a$ for the topological structure in question.'
%{\sf the precise meaning to the phrase
%``whenever $x$ is sufficiently near $a$, $x$ has the property $P\{x\}$}''
%is that $\bigcup_{a\in X}\{a\}\times P_a(x)$ is topoic.
%to be ``$P$ is topoic'' and introduce the notion of a topoic structure.
%%We shall say that a set $E$ carries a {\em topoic structure} whenever we have
%%defined, by some means or other, a family
%%of subsets of $E^n$, $n>0$, which are called {\em topoic subsets of $E^n$} - provided
%%that these topoic subsets satisfy certain conditions (the axioms
%%of topoic structures). These conditions are such that a topoic structure
%%determines a simplicial object in the categories of filters,
%%i.e.~a functor $$ \Ord{\omega}^{op} \xra{\ttt(X)} \Filt.$$
%%
\vskip 3pt
{\sf As we have already said,} talking about topoic subsets wrt {\sf a topological structure on a set enables one
to give an exact meaning to the phrase ``whenever $x$ is sufficiently
near $a$, $x$ has the property $P\{x\}$''. But, apart from the situation
in which a ``distance'' has been defined, it is not clear
what meaning
ought to be given to the phrase ``every pair of points $x,y$ which are sufficiently near each other has
the property $P \{x, y\}$'', since {\em a priori}
} topoic subsets wrt a topological structure give {\sf no means of comparing the neighbourhoods of two different
points. Now the notion of a pair of points near to each other arises fre%-
quently in classical analysis (for example, in propositions which involve
uniform continuity). It is therefore important that we should be able
to give a precise meaning to this notion in full generality, and we are
thus led to define} the notion of topoic subsets wrt {\sf structures which are richer than topological structures,
namely {\em uniform structures}.}
%Similarly, {\sf in the situation in which a ``distance'' has been defined},
For a metric space $M$,
call a subset $P\subseteq M\times M$ {\em topoic} iff
%to the phrase
``{\sf every pair of points $x,y$ which are sufficiently near each other has
the property $P \{x, y\}$}'', i.e.~there is $\varepsilon>0$ such that
$(x,y)\in P$ whenever $dist(x,y)<\varepsilon$. More generally,
call a subset $P\subseteq M^n$ {\em topoic} iff
there is $\varepsilon>0$ such that
$(x_1,...,x_n)\in P$ whenever $dist(x_i,x_j)<\varepsilon$
for each $1\leq i\leq j\leq n$.
The axioms of uniform structure, cf.~\href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki,II\S I.1]},
say that the filters of topoic subsets on $M^n$, $n\geq 0$, define a
2-dimensional simplicial object %functor
$$
\mU(M): \Ord{\omega}^{op} \lra \Filt$$
which factors as
$$\Ord{\omega}^{op} \lra FinSets^{op} \lra \Filt$$
where $FinSets^{op}$ is the category of finite sets.
%%(V I) Every set belonging to U contains the diagonal d.
%%-1
%%(VII) If V e U then V e U.
%%(VIII) For each V e U there exists W e U such that WoW c V.
%%
In particular,
to say the map of filters $M\xra{(x,x)} M\times M$ is continuous and the filter on $M$ is antidicrete
is almost to say Axiom ($\mathrm U_{\mathrm {I}}$) {\sf Every set belonging to} the set of
entourages $ \mathfrak U$ {\sf contains the diagonal $\Delta$} of $M\times M$.
Axiom ($\mathrm U_{\mathrm {II}}$) {\sf If $V \in \mathfrak U$ then $V^{-1}\in \mathfrak U$},
where $V^{-1}=\{(y,x):(x,y)\in V\}$, says the permutation of coordinates $M\times M\xra{(x,y)\mapsto (y,x)} M\times M$
is continuous.
Axiom ($\mathrm U_{\mathrm {III}}$) {\sf For each} entourage {\sf $V\in \mathfrak U$ there exists} entourage {\sf $ W \in \mathfrak U$
such that $W\circ W \subset V$} says that
$pr_{1,3}:M\times M\times M\lra M\times M, \ (x_1,x_2,x_3)\mapsto (x_1,x_3)$
is continuous wrt the filter on $M\times M\times M$ defined as
pullback along projections $M\times M \xleftarrow {pr_{1,2}}M\times M\times M\xra{pr_{2,3}}M\times M$.
Thus Axiom ($\mathrm U_{\mathrm {I}}$) and ($\mathrm U_{\mathrm {III}}$) of uniform structures say
the functor $
\mU(M): \Ord{\omega}^{op} \lra \Filt$ is well-defined, and Axiom ($\mathrm U_{\mathrm {II}}$)
says it factors via $\Ord{\omega}^{op} \lra FinSets^{op}$.
This terminology gives us {\sf means of comparing the neighbourhoods of two different
points} and {\sf give a precise meaning to the notion of a pair of points near to each other}
which {\sf arises frequently 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 axioms of uniform structure say
%%that the filters of topoic subsets on $M^n$, $n\geq 0$, define a functor
%%$$
%%\mU(M): \Ord{\omega}^{op} \lra \Filt$$
%%which factors as
%%$$\Ord{\omega}^{op} \lra FinSets^{op} \lra \Filt$$
%%where $FinSets^{op}$ is the category of finite sets.
%%
% This gives an example of a topoic structure
%on the set of points of a metric space $M$.
% we give {\sf the precise meaning} $P$ is topoic
%with respect to a certain %the
%topoic structure associated to the distance (metric) on a set $E$.
A topological argument often relies on consequently choosing ``sufficiently near''
points; in this case we expect that
it implicitly constructs a topoic subset of $E\times .. \times E$.
Sometimes an argument chooses points not consequently, and we hope that often enough
it implicitly constructs a topoic subset of $E\times .. \times E$,
albeit in a topoic structure not associated with a topological structure
and possibly specific to the argument.
%%Evidently the choice of axioms to be imposed
%%is to some extent arbitrary, but also depend on whether we consider
%%topoic structures associated with topological or metric structures.
%%In this note we do not discuss this choice.
%%
%%The conditions on topoic structures associated with a topological structure
%%will enable us to define a simplicial object in the category of filters,
%%i.e.~a functor
%%$$\ttt(X): \Ord{\omega}^{op} \lra \Filt$$
%%where
%%$\Filt$ is the category of filters, and
%% $\Ord{\omega}$ denotes the category of categories corresponding to finite linear orders
%%$$\bullet_1\lra .. \lra \bullet_n,\ 0\leq n <\omega.$$
%%
%%The conditions on topoic structures associated with a metric structure
%%will enable us to define in \S\ref{met:filt} a simplicial object in the category of filters,
%%a functor
%%$$\mU(M): \Ord{\omega}^{op} \lra \Filt$$
%%which factors as
%%$$\Ord{\omega}^{op} \lra FinSets^{op} \lra \Filt$$
%%where $FinSets^{op}$ is the category of finite sets.
%%
%%Thereby we shall obtain fully faithful embeddings of the category of metric spaces (uniform spaces)
%%and uniformly continuous maps
%%and that of topological spaces
%%in the category of simplicial filters
%%$$\ttt: Top \subset Func( \Ord{\omega}^{op} , \Filt)$$
%%$$\mU: UniformSpaces \subset Func( \Ord{\omega}^{op} , \Filt)$$
%%
%%
Let us now define the %a
topoic structure on a set $E$
associated with a topological structure on $E$.
\subsubsection{\label{def:topoic-top}Topoic structure of a topological space.}
Let $X$ be a topological space. Call a property (subset) $P\subseteq X\times X$ {\em topoic} iff
$(a,x)\in P$ holds whenever $x$ is sufficiently near $a$, i.e.
for each point $a\in X$ there is a neighbourhood $U_a$ such that $(a,x)\in P$ whenever $x\in U_a$.
Call a property $P\subseteq X^n$ {\em topoic} iff
we can ensure that $(x_1,..,x_n)\in P$ provided
%>>$x_{i+1}$ is sufficiently near $x_i$, for $0< i0$ such that $(x_1,..,x_n)\in M$ provided
$dist(x_i,x_j)<\varepsilon$ for each $1\leq i\leq j\leq n$. Thereby we give
the phrase ``{\sf every pair of points $x,y$ which are sufficiently near each other
has the property $P \{x, y\}$}'' the precise meaning that $P$ is topoic with respect to
the topoic structure associated with the metric (distance) on $M$.
%%+ we have no means of comparing the neighbourhoods of two different
%Call a property (subset) $P\subseteq M\times M$ {\em topoic} iff
%$(x,y)\in P$ holds whenever $x$ is sufficiently near $y$, i.e.~for some $\varepsilon>0$,
%$(x,y)\in P$ provided $dist(x,y)<\varepsilon$.
%Call a property $P\subseteq M^n$ {\em topoic} iff
%we can ensure that $(x_1,..,x_n)\in P$ provided
%>>$x_{i+1}$ is sufficiently near $x_i$, for $0< i0$.
Given a mapping $f:M\lra M'$ of sets of points, the condition that
the preimage of a topoic subset of $M\times M$ is necessarily a topoic
subset of $M'\times M'$,
says that for each $\delta>0$ there is $\varepsilon>0$ such that
$\dist(f(x),f(y))<\delta$ whenever $\dist(x,y)<\varepsilon$, i.e.
the mapping $f$ is uniformly continuous.
In fact, as is easy to see, this construction also works for uniform spaces,
and we obtain a fully faithful embedding of the category of uniform spaces
%metric spaces and uniformly continuous maps
in the category of simplicial filters\footnote{
For more details see
\href{http://mishap.sdf.org/mints/mints_simplicial_filters.pdf}
{[Gavrilovich, Simplicial Filters]}, in particular Claim~2 which characterises the category of functors
corresponding to uniform spaces.
}
$$\mU: UniformSpaces \subset Func( \Ord{\omega}^{op} , \Filt)$$
For a filter $\mathfrak F$, let $\iembE(\FFF)=Hom(-,\FFF)$ denote the simplicial filter
$(\FFF, \FFF\times \FFF, \FFF\times \FFF\times \FFF, ...)$
consisting of Cartesian powers of $\FF$ and coordinate maps.
Let $\iemb(\FFF)$ denote the simplicial filter
$(\FFF, \FFF, \FFF, ...)$
consisting of $\FF$ itself and identity maps.
A {\em Cauchy filter} $\FFF$ on a metric space $M$ (cf.~\href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki,II\S3.1,Def.2]})
is a filter on the set of points of $M$ such that
%one of the two equivalent conditions holds:
%\bi\item for each $\varepsilon>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 $ \iembE(\FFF) \lra \mU (M)$ is well-defined.
%\ei
A {\em Cauchy sequence} in $M$ is a map $\iembE(\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).
This allows to define various notions of equicontinuity of sequences of functions.
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 ${\varepsilon > 0}$,
there exists a neighbourhood ${U}$ of ${x}$ such that
${d_Y(f_i(x'), f_i(x)) \leq \varepsilon}$ 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 ${\varepsilon > 0}$ there exists a ${\delta > 0}$ such that
${d_Y(f_i(x'), f_i(x)) \leq \varepsilon}$ 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 ${\varepsilon > 0}$ there exists a ${\delta > 0}$ and $N>0$ such that
${d_Y(f_i(x'), f_j(x)) \leq \varepsilon}$ for all $i,j>N$ and ${x', x \in x}$ with ${d_X(x,x') \leq \delta}$.
\item the map $\mU(X)\times \iembE(\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$.
This suggest we might reformulate Arzela-Ascoli theorem as something about inner Hom in $\sFilt$,
see Question~\ref{q:ascoli}.
\subsubsection{\label{measure:filt}Measure spaces.}\footnote{This section is not finished, in a very preliminary state and may contain mistakes.
I will appreciate any corretions and suggestions sent by readers.}
We can associate a topoic structure on a set with a measure on the set
as follows.
Let $X$ be a set and $\mu$ be a measure on $X$.
We say a subset $U$ of $X^n$ is {\em $\mmu_{\!\!\!<\omega}$-big} iff there exists finitely many subsets $A_1,..,A_N$ such that
$\mu(X\setminus \cup_{1\leq i\leq N} A_i)=0$ and $\cup_{1\leq i \leq N} A_i^n\subset U$.
We say a subset $U$ of $X^n$ is {\em $\mmu$-big} iff there exists countably many subsets $A_1,A_2,..$ such that
$\mu(X\setminus \cup_{1\leq i< \omega} A_i)=0$ and $\cup_{1\leq i < \omega} A_i^n\subset U$.
Similarly to above, these filters on the Cartesian powers of $X$ and the coordinate maps
define simplicial objects in the category of filters
$$(X_{\mmufini},(X\times X)_{\mmufini}, (X\times X\times X)_{\mmufini}, ...)$$
$$(X_{\mmu},(X\times X)_\mmu, (X\times X\times X)_\mmu, ...)$$
which we denote by $\mmufini(X,\mu)$ and $\mmu(X,\mu)$, resp.
In fact these functors factor through the category of finite sets:
$$\mmufini(X,\mu):\Ord{\omega}^{op} \lra FinSets^{op} \lra \Filt$$
$$\mmu(X,\mu):\Ord{\omega}^{op} \lra FinSets^{op} \lra \Filt$$
Let $Y$ be a set and $\nu$ be a measure on $Y$.
A measurable map $f:X\lra Y$ such that $\mu(f\inv(A))=0$ whenever $\nu(A)=0$
induces morphisms of functors $\mmufini(X,\mu) \xra{\mmufini(f)} \mmufini(Y,\nu)$ and $\mmu(X,\mu) \xra{\mmu(f)} \mmu(Y,\nu)$,
and conversely, each morphism in $\sFilt$ between these objects is of this form.
\subsubsection{\label{geodesic:filt}Maps of metric spaces preserving geodesics.}\footnote{This section is not finished, in a very preliminary state and may contain mistakes. I will appreciate any corrections and suggestions sent by readers.}
%%We say that a map $f:M\lra M'$ of metric spaces {\em preserves geodesics}
%%iff the image $f\circ\gamma:[0,1]\lra M'$ of a geodesic curve $\gamma:[0,1]\lra M$
%%is geodesic.
%%
For a metric space $M$, call a subset $P\subset M^n$ {\em topoic wrt geodesic structure} iff
$(x_1,...,x_n) \in P$ whenever
\begin{itemize}
\item[$(*)$] there is a geodesic in $M$ first passing through $x_1$,
then passing through $x_2$, ..., then passing through $x_n$.
\end{itemize}
The condition $(*)$ is preserved by coordinate maps $M^n\lra M^m$
preserving the order of coordinates, hence
this does define a simplicial filter based on $(M,M^2,M^3,...)$.
A map $f:M\lra M'$ induces a map of these topoic structures on $M$ and $M'$
if a geodesic in $M$ maps into a geodesic in $M'$.
The converse holds for $M$ and $M'$ nice enough, e.g.~Riemannian manifolds
where geodesics are locally unique.
Note that the topoic structure on $M^2$ is trivial if
each pair of points on $M$ can be connected by a geodesic.
%Probably this construction can be modified
%in many obvious ways (take shortest paths
%instead of geodesics (which are only locally shortest);
%take geodetics up to some \epsilon).
%
Note that this definition can be modified in some obvious ways,
e.g.~call a subset $P\subset M^n$ {\em topoic wrt $+\varepsilon$-geodesic
structure}, resp.~{\em $\cdot\varepsilon$-geodesic structure},
iff there is $\varepsilon>0$ such that
$(x_1,...,x_n)\in P$ whenever $(*)_{+\varepsilon}$, resp.~$(*)_{\cdot\varepsilon}$, holds:
\begin{itemize}
\item[$(*)_{+\varepsilon}$] for any $1\leq iM' of metric spaces is {quasi-symmetric}
%%iff either of the following equivalent conditions holds:
%%
%%(a) for each alpha>0 there is beta>0 such that
%%for any different points x,y,z of M
%%
%%dist(x,y)/dist(y,z) < alpha implies
%%dist(f(x),f(y))/dist(f(y),f(z)) < beta
%%
%%
%%(b) there exists a monotone function \nu: [0,\infty)--->[0,\infty)
%%such that
%%for any dneefrfit ponits x,y,z of M
%%
%%dist(f(x),f(y))/dsit(f(y),f(z)) < \nu( dist(x,y)/dist(y,z) )
%%
%%
%%
%%Call a subset P of M^n {topoic} iff there is alpha>0 such that
%%
%%(x_1,..,x_n) \in P whenever for each 0*M' of metric spaces induces
%%a functor between teh corresponding simplicial filters
%%iff f is quasi-symmetric.
%%
%%
%%
%%
%%A map $f:M\lra M'$ of metric spaces is called {\em quasi-symmetric}
%%iff there is a monotone continious function $\eta:[0,\infty)\lra [0,\infty)$
%%such that %either of the following equivalent conditions holds:
%%for any $t\in \RR$, $x,y,z\in M$ it holds
%%$$\dist(y,z)0$ there is $\beta>0$ such that
%%for any different points $x,y,z \in M$
%%it holds:
%%$$
%%\frac{\dist(x,y)}{\dist(y,z)} < \alpha \implies
%%\frac{\dist(f(x),f(y))}{\dist(f(y),f(z))} < \beta$$
%%
%%Call a subset $P$ of $M^n$ {\em topoic} iff there is $\alpha>0$
%%such that
%%$(x_1,..,x_n) \in P$ whenever for each $1\leq i}[ur]|{n\mapsto (a,a_n)} \ar[r]_{a\mapsto a_n} & {\RR_{\mathrm{antidiscrete}\!\!\!\!\!\!\!\!\!\!\!\!\!\!}}
}$$
Simplicially, map $\RR\times \RR_{\ttt}\lra \RR_{\mathrm{antidiscrete}}$ is part of the simplicial map
$\ttt(\RR)_+\lra \ttt(\RR)$ forgetting the first face and degeneracy maps; here $ \ttt(\RR)$ denotes
the simplicial object corresponding to $\RR$ and $\ttt(\RR)_+$ denotes the shifted simplicial object $(\RR\times \RR\,_\ttt, \RR\times \RR\times \RR\,_\ttt,...)$. The object $\ttt(\RR)_+$ is a disconnected union of fibres above points of $\RR$, and hence
any map from a connected simplical object is necessarily constant on the first coordinate.
This suggests that we rewrite the diagram above as the following lifting diagram of simplicial objects:
$$
\xymatrix
{ {} & {} \\
{} & {} \\
{} & {\ttt(\RR)_+}\ar[d]|{pr_{2,3,...}}\\
{E_\diag(\NN)}\ar@{-->}[ur]\ar[r] & {\ttt(\RR)}
}
\xymatrix{{}&{}}
\xymatrix
{{...} \ar[d]\ar@/_1pc/[rr]& {...} \ar[d] \ar[dr]|{pr_{2,..}} & {...}\ar[d] \\
{\NN\times \NN\times \NN_{\diag \NN}}\ar[d]\ar@/_1pc/[rr] \ar@{-->}[ur] & {\RR\times \RR\times \RR_\ttt}\ar[d] \ar[dr]|{pr_{2,3}} & {\RR\times \RR\times \RR_\ttt} \ar[d] \\
{\NN\times \NN_{\diag \NN}}\ar[d]\ar@/_1pc/[rr] \ar@{-->}[ur] & {\RR\times \RR_\ttt} \ar[dr]|{pr_2} & {\RR\times \RR_\ttt} \ar[d] \\
{\NN}\ar[rr] \ar@{-->}[ur] & {} & {\RR}
}
$$
The filters on Cartesian powers of $\NN$ are defined to be the finest possible so that continuity does not place unnecessary restrictions,
i.e.~the filter $\diag(\NN)$ on $\NN\times ...\times \NN$ is the finest possible such that the diagonal embedding $\NN\lra \NN\times ... \times \NN $ is continuous,
i.e. a subset of $\NN^n$ is big iff it contains the image of a big(=cofinite) subset of $\NN$.
{\sf This is a general
fact:}
\begin{enonce}{Reformulation}
{\sf whenever we speak of limit, we are considering} %a mapping $f$ of a
a lifting diagram
$$
\xymatrix
{ {} & {} \\
{} & {} \\
{} & {\ttt(X)_+}\ar[d]|{pr_{2,3,...}}\\
{E_\diag(F)}\ar@{-->}[ur]\ar[r] & {\ttt(X)}
}$$
where $F$ is a filter and $X$ is a topological space, and
the map $\ttt(X)_+\lra \ttt(X)$ is the simplicial map forgetting the first face and degeneracy maps,
and $E_\diag:\Filt\lra \sFilt$ as defined above.
\end{enonce}
This allows us to formulate several properties of spaces as lifting properties
(here $\bot$ and $\top$ denote the initial and terminal object of $\sFilt$):
\bi\item
a topological space $K$ is quasi-compact iff each ultrafilter $\mathcal U$ on $K$ converges, i.e.,
$$\bot\lra E_\diag(\mathcal U) \rtt \ttt(K)_+\lra\ttt(K)$$
\item a metric space $M$ is quasi-compact iff
each ultrafilter $\mathcal U$ on $M$ converges %i.e.
$$\bot\lra E_\diag(\mathcal U) \rtt \mU(M)_+\lra\mU(M)$$
\item a metric space $M$ is complete iff
each Cauchy filter $\mathcal F$ on $M$ converges%, i.e.
$$\bot\lra E(\mathcal F) \rtt \mU(M)_+\lra\mU(M)$$
\ei
Also note
\bi
\item a metric space $M$ is pre-compact iff
each ultrafilter $\mathcal U$ on $M$ is Cauchy%, i.e.
$$ \mathrm{const}\,\mathcal U \lra E(\mathcal U) \rtt \mU(M)\lra\top $$
\ei
%set $E$ into a topological space $F$,} i.e. $f:E\lra F$, {\sf and we say that $f$ has a point $a$ of $F$
%as a limit if} $$f_a:E\lra F\times F_\ttt, \ \ x \longmapsto (a,f(x)) $$ is continous
%wrt to topology on $E$ formed by {\sf a certain family $\mathfrak F$ of subsets
%of $E$, given beforehand.} Expanding the definitions gives
%that~{\sf the set of elements $x$ of $E$ whose image $f (x)$ belongs to
%a neighbourhood $V$ of $a$ [this set is just the ''inverse image'' $f^{-1} (V)$]
%belongs, whatever the neighbourhood $V$, to} family $\mathfrak F$.
%a certain family $\mathfrak F$ of subsets of $E$, given beforehand.}
%For the notion of limit to have the essential
%properties ordinarily attributed to it, the family
% must satisfy certain
%axioms, which are stated in Chapter I,
% 6. Such a family
% of subsets
%of E is called afilter on E. The notion of a filter, which is thus inseparable
%from that of a limit, appears also in other contexts in topology; for example,
%the neighbourhoods of a point in a topological space form a filter.
%The general study of all these notions is the essential purpose of Chapter I.
%
%%
%%
%%
%%So we consider a mapping $f:E\lra F$ between two sets, {\sf each endowed with suitable structures}
%%described rather explicitly: $F$ {\sf is assumed to carry a topological structure},
%%%it consists of {\sf neighbourhood[s] V of $a$} where $a\in F$ is some fixed point.
%%The {\sf suitable structure} on $E$ consists of {\sf a certain family of
%%subsets, given beforehand} which forms a filter, which we will denote $\FFF$.
%%The condition on $f$ says that {\sf ``the inverse image'' $f^{-1}(V)$ belongs
%%to} the filter, {\sf whatever the neighbourhood $V$} of $a\in F$.
%%That is, the mapping $f:E\lra F$ is continuous with respect to the neighbourhood
%%filter $\mathfrak B(a)$ of point $a\in F$ on $F$
%%and the given filter $\FFF$ on $E$.
%%%---
%%%---
%%%--- This mentions a filter $\mathfrak F$ on a set $E$, and
%%%--- %a filter on the set of points of a $Y$ which is
%%%--- the neighbourhood filter $\mathfrak B(a)$ of a point $a\in Y$;
%%%--- recall that the set of all neighbourhoods
%%%--- of a point $a$ in $Y$ is a filter, called the neighbourhood filter of $a\in Y$.
%%%--- The function $f:E\lra Y$ is
%%%--- required to be continuous with respect to these filters, i.e. we have an arrow
%%%--- $$f:E_{\mathfrak F}\lra Y_{\mathfrak B(a)}$$
%%%--- where by $E_{\mathfrak F}$, resp.~$ Y_{\mathfrak B(a)}$ we denote the sets $E$, resp.~$Y$,
%%%--- equipped with the corresponding filters.
%README:
%TODO: note this sentence was not well-justified! and indeed, we missed the right formulation because of this...
%%-- We'd like to have a simplicial map, and indeed $f_a$ extends to
%%-- $$
%%-- \xymatrix{ {} & {E_{\mathfrak F}} \ar[r]|{\text{id}} \ar@{->}[d]|{x\mapsto(a,f(x))} & {E_{\mathfrak F}} \ar[r]|{\text{id}}\ar[l] \ar[d]|{x\mapsto(a,f(x),f(x))} & {E_{\mathfrak F}} \ar[r]|{\text{id}}\ar[l] \ar[d]|{x\mapsto(a,f(x),f(x),f(x))} & {} \ar[l] \\
%%-- {} &{ {\{a\}\times F}_{\ttt} } \ar[r]\ar[d] & { {\{a\}\times F\times F}_\ttt } \ar[r]\ar[l]\ar[d] & { {\{a\}\times F\times F\times F}_\ttt } \ar[r]\ar[l]\ar[d] & {} \ar[l] \\
%%-- {F} \ar[r] &{ {F \times F}_{\ttt} } \ar[r]\ar[l]\ar[d]|{(x_1,x_2)\mapsto x_2} & { {F \times F\times F}_\ttt } \ar[r]\ar[l]\ar[d]|{(x_1,x_2,x_3)\mapsto (x_2,x_3)} & { {F \times F\times F\times F}_\ttt } \ar[r]\ar[l]\ar[d]|{(x_1,x_2,x_3.x_4)\mapsto (x_2,x_3.x_4)} & {} \ar[l] \\
%%-- {} & {F} \ar[r] &{ {F \times F}_{\ttt} } \ar[r]\ar[l] & { {F \times F\times F}_\ttt } \ar[r]\ar[l] %& { {F \times F\times F\times F}_\ttt } \ar[r]\ar[l]
%%-- & {} \ar[l]
%%-- }
%%-- $$
%%-- Here subscript ${}_\ttt$ means that we consider sets endowed with the filter of
%%-- topoic subsets associated with the topological space $F$.
%%--
%%-- Let us ponder further; the diagram implicitly uses infinitary notation $x\mapsto(a,f(x),f(x),...,f(x))$ %is implicitly infinite
%%-- and thus
%%-- somewhat unsatisfactory for us.
%%-- Denote $X=\{a\}\times F$ and rewrite the diagram as
%%-- $$
%%-- \xymatrix{ {} & {E_{\mathfrak F}} \ar[r]|{\text{id}} \ar@{->}[d] & {E_{\mathfrak F}} \ar[r]|{\text{id}}\ar[l] \ar[d] & {E_{\mathfrak F}} \ar[r]|{\text{id}}\ar[l] \ar[d] & {} \ar[l] \\
%%-- {} &{X_{\mathfrak B} } \ar[r]\ar[d]|{x\mapsto(a,f'(x))} & { {X\times X}_{\mathfrak B} } \ar[r]\ar[l]\ar[d]|{x\mapsto(a,f'(x),f'(x))} & { { X\times X\times X}_{\mathfrak B} } \ar[r]\ar[l]\ar[d]|{x\mapsto(a,f'(x),f'(x),f'(x))} & {} \ar[l] \\
%%-- {F} \ar[r] &{ {F \times F}_{\ttt} } \ar[r]\ar[l]\ar[d]|{(x_1,x_2)\mapsto x_2} & { {F \times F\times F}_\ttt } \ar[r]\ar[l]\ar[d]|{(x_1,x_2,x_3)\mapsto (x_2,x_3)} & { {F \times F\times F\times F}_\ttt } \ar[r]\ar[l]\ar[d]|{(x_1,x_2,x_3.x_4)\mapsto (x_2,x_3.x_4)} & {} \ar[l] \\
%%-- {} & {F} \ar[r] &{ {F \times F}_{\ttt} } \ar[r]\ar[l] & { {F \times F\times F}_\ttt } \ar[r]\ar[l] %& { {F \times F\times F\times F}_\ttt } \ar[r]\ar[l]
%%-- & {} \ar[l]
%%-- }
%%-- $$
%%--
%%-- Finally, note that the underlying sets in the second row form the simplicial object of Cartesian powers $(X,X^2,X^3,...)$,
%%-- and that any map from such an object to the third row is necessary of form\footnote{
%%-- The following easy argument shows this.
%%-- Let $\mathcal X$ and $\mathcal Y$ be objects of $\sFilt$ such that are setwise objects of Cartesian powers of sets $X$ and $Y$ and coordinate maps.
%%-- Then a morphism $\mathcal X\circ [0]\lra \mathcal Y$ determines a point of $a\in Y$ and is necessarily of form
%%-- $(x_1,..,x_n)\mapsto (a,f(x_1),..,f(x_n))$ for some function $f:X\lra Y$. To see this, take any two points $x_1,x_2\in X$
%%-- and consider the coordinate projections $(a,y_1)=f(x_1)$ and $(a,y_2)=f(x_2)$
%%-- of $(a,y_1,y_2)=f(x_1,x_2)\in Y\times Y\times Y$ on the second and third
%%-- coordinate.
%%-- } indicated in the diagram,
%%-- i.e.~$X^n\lra F^{n+1},\ (x_1,...,x_n)\mapsto (a,f'(x_1),...,f'(x_n))$
%%-- for some function $f':X\lra F$.
%%--
%%-- \begin{enonce}{Reformulation}
%%-- whenever we speak of limit, we are considering a factorisation of a map $f:\iemb(E)\lra \ttt(F)$ of simplicial filters
%%-- as $$
%%-- \iemb(E) \xdashrightarrow{\ \ \ \ } X \xdashrightarrow{\ \ \ \ } \ttt(F)[0] \lra \ttt(F)
%%-- $$
%%-- via an object which is set-wise the object of Cartesian powers of a set,
%%-- and the ``shift'' of the topological space ``forgetting'' the first coordinate.
%%-- \end{enonce}
%%--
%%-- Let us now introduce notation to describe this in detail.
%%--
%%-- For a filter $\FFF$, let $\iemb(\mathfrak F)$ denote the constant simplicial object
%%-- defined by $\iemb(\FFF)(n)=\FFF$ and $\iemb(\FFF)(f)=\id$ for any morphism $f$.
%%--
%%--
%%--
%%-- Let $[0]:\Ord{\omega}^{op}\lra \Ord{\omega}^{op}$ denote the endofunctor
%%-- appending a new least element to each linear order, i.e.
%%-- $$1< ... < n \,\, \longmapsto\,\, 0 < 1 <...}[rd]|{(wf)} & {} \\
&A\ar[rr] & {} & B}$
$\xymatrix{
& {} & A\times_B B^I \ar@{<--}[ld]|{(cw)}\ar@{-->}[rd]|{(f)} & {} \\
&A\ar[rr] & {} & B}$
Figure 3 gives drawings
representing these decompositions in the category of topological spaces.
\newline\noindent\includegraphics[width=\linewidth]{M2-pix-with-LS.png}
\subsubsection{($cw$)($f$)-decomposition.}
Let us analyse Figure~3b (cw-f decomposition).
Recall that, to translate, we care both about intuition
and algebraic manipulations.
The construction uses the following algebraic manipulations:
it considers pairs $(x, \gamma^{const}_{f(x)})$ and
$(x,\gamma(t_1))$. We ignore paths because these are complicated
``infinitary'' notions we are unable to express in our language,
and hence all we are left with are pairs $(x,f(x))$ where $ x\in A$ and
$(x,y)$ where $x\in A,y\in B$.
This suggests we look at the following decomposition:
\newline\noindent
$
\xymatrix{
{X} \ar[r]\ar[d]|{(x,f(x))} &{ X\times X } \ar[r]\ar[l]\ar[d]|{(x_1,f(x_1),x_2,f(x_2))} & { X\times X\times X } \ar[r]\ar[l]\ar[d]|{(x_1,f(x_1),x_2,f(x_2),x_3,f(x_3))} & {} \ar[l] \\
{X\times Y} \ar[r]\ar[d]|{y} &{ X\times Y\times X\times Y } \ar[r]\ar[l]\ar[d]|{(y_1,y_2)} & { X\times Y\times X\times Y\times X\times Y } \ar[r]\ar[l]\ar[d]|{(y_1,y_2,y_3)} & {} \ar[l] \\
{ Y} \ar[r] &{ Y\times Y } \ar[r]\ar[l] & { Y\times Y\times Y } \ar[r]\ar[l] & {} \ar[l] } \\
$
What is the filter $\mathfrak F_{\text{cw-f}}$ on the elements of the middle row $X\times Y\times X\times Y \times ...$?
The diagram suggests that we start with the filter corresponding to the product topology on $X\times Y$.
Let us use the intuition. In model categories,
weak equivalences are thought of as equivalences and therefore
$X$ and $(X\times Y)_{\mathfrak F_{\text{cw-f}}}$ should be very similar for purposes we care about.
Geometric intuition suggests that we only care about an infinitesimal neighbourhood of
$\{(x,id_x):x\in X\}$ and would prefer our paths to be infinitesimally short.
This motivates us to modify the topoic filter of the product topology on $X\times Y$ by adding as topoic
``infinitesimal neighbourhoods of $X$''
$$\bigcup\limits_{x\in X,\, f(x)\in U_{f(x)}\text{ a neighbourhood}}\,\{x\}\times U_{f(x)} \subset X\times Y$$
That is, we define a new filter $\mathfrak F_{\text{cw-f}}^{n}$ on $(X\times Y)^n$ generated by subsets
%by saying that a subset $V\subset (X\times Y)^n$ is {\em topoic} iff it is of form
$U^n\cap W$ where $U$ is of the form above and $W$ is a topoic subset of $(X\times Y)^n$
with respect to the product topology on $X\times Y$.
\begin{remark} Note that when $X=a$ is a point, this decomposition gives us
$\{a\}\xra{(cw)}Y\circ[0]_a\xra{(f)} Y$.
This suggests that we think of $Y\circ[0]_a$ as the space of infinitesimally short paths starting at $a$;
this would correspond to the intuition that an infinitesimally short path is roughly the same as its endpoint.
\end{remark}
%%%--- It helps to think directly in the intuitive terms:
%%%--- our situation (topology on $X$, on $Y$ and a continous function $X \xra f Y$)
%%%--- enables us to give {\em precise meaning} to the following three operations:
%%%--- to pick an $x'\in X$ near $x\in X$, to pick an $y'\in Y$ near $y\in Y$, and
%%%--- to pick an $y'\in Y$ near $f(x), x\in X$.
%%%--- A filter on $X\times Y\times X\times Y \times ...$ would allow us to give
%%%--- precise meaning to a {\em sequence} of these operations.
%%%--- The product topology on $X\times Y$ let us give
%%%--- precise meaning to the following: %pick $x_1\in X, y_1\in Y$, then
%%%--- {\em simultaniously} pick $x_2\in X$ near $x_1$ and $y_2\in Y$ near $y_1$
%%%--- (that is, pick a neighbourhood $U_{x_1}\times V_{y_1}\ni (x_1,y_1)$),
%%%--- then {\em simultaniously} pick $x_3\in X$ near $x_2$ and $y_3\in Y$ near $y_2$,...
%%%---
%%%---
%%%---
%%%--- ..The product topology on $X\times Y$ let us give
%%%--- precise meaning to the following:
%%%--- pick $y_1$ near $f(x_1)$, then
%%%--- {\em simultaniously} pick $x_2\in X$ near $x_1$ and $y_2\in Y$ near $y_1$
%%%--- (that is, pick a neighbourhood $U_{x_1}\times V_{y_1}\ni (x_1,y_1)$),
%%%--- then pick >>$y_2\in Y$ near $f(x_2)$, ??
%%%---
%%%---
%%%--- ...It reminds of the product of $X$ and $Y$, and this leads
%%%--- a subset of $(X \times Y)^n$ is {\em topoiq} iff
%%%---
%%%--- Call a property $P\subseteq (X\times Y)^n$ {\em topoic} iff
%%%--- we can ensure $(x_1,y_1,..,y_n,x_n)\in P$
%%%--- by consequtively picking $y_{1}$ sufficiently near $f(x_1)$,
%%%--- then picking $x_2$ sufficiently near $x_i$,
%%%--- then picking $y_2$ sufficiently near $f(x_2)$, OR near $y_1$!!!!!
%%%--- then picking $x_3$ sufficiently near $x_2$,....
%%%--- Put more formally,\\
%%%--- \noindent\begin{small} Q: what depends on what???
%%%--- for each $x_1\in X$ there is an open neighbourhood $U_{x_1}\ni x_1$ and
%%%--- there is a neighbourhood $V_{x_1}\ni f(x_1)$ such that\\
%%%--- for each $y_1\in V_{x_1}$ there is a neighbourhood $U_{x_1,y_1}\ni y_1$ such that\\
%%%--- for each $x_2\in U_{x_1,y_1}$ there is a neighbourhood $V_{x_1,y_1,x_2}\ni f(x_2)$ such that\\
%%%--- for each $y_2\in V_{x_1,y_1,x_2} \cap V_{x_1,y_1,x_2} $ there is a neighbourhood $U_{x_1,y_1,x_2,y_2}\ni x_3$ such that\\
%%%--- for each $x_3\in U_{x_1,y_1,x_2,y_2}$ there is a neighbourhood $V_{x_1,y_1,x_2,y_2.x_3}\ni f(x_3)$ such that\\
%%%--- for each $y_3\in V_{x_1,y_1,x_2,y_2.x_3}$ there is a neighbourhood $V_{x_1,y_1,x_2,y_2.x_3,y_3}\ni x_4$ such that\\
%%%--- .....\\
%%%--- %for each point $x_n\in U_{x_1,x_2,...,x_{n-1}}$ there is a neighbourhood $ U_{x_1,x_2,...,x_{n-1}}\ni x_n$ such that\\
%%%--- $(x_1,y_1,...,x_n,y_n)\in P$. \\
%%%--- \end{small}
%%%---
%%%---
%%%--- Let us use the intuition. In model categories,
%%%--- weak equivalences are thought of as equivalences and therefore
%%%--- $A$ and $A'$ should be very similar for purposes we care about.
%%%--- Geometric intuition suggests that we only care about an infinitesimal neighbourhood of
%%%--- $\{(x,id_x):x\in X\}$ and would prefer our paths to be infinitesimally short.
%%%---
%%%--- This motivates us to modify the topoic filter on $X\times Y$ by adding as topoic
%%%--- ``infinitesimal neighbourhoods of $X$''
%%%--- $$\bigcup\limits_{x\in X,\, f(x)\in U_{f(x)}\text{ a neighbourhood}}\,\{x\}\times U_{f(x)} \subset X\times Y$$
%%%--- That is, we define a new filter on $(X\times Y)^n$ generated by subsets
%%%--- %by saying that a subset $V\subset (X\times Y)^n$ is {\em topoic} iff it is of form
%%%--- $U^n\cap W$ where $U$ is of the form above and $W$ is a topoic subset of $(X\times Y)^n$
%%%--- with respect to the product topology on $X\times Y$.
%%%---
%%%---
%%%--- \begin{remark} Note that when $X=a$ is a point, this decomposition gives us
%%%--- a map $\{a\}\xra{(cw)}Y[0]_a\xra{f} Y$.
%%%--- This suggests that we think of $Y[0]_a$ as the space of infinitesimally short paths starting at $a$;
%%%--- this would correspond to the intution that an infinitisemally short path is roughly the same as its endpoint.
%%%--- \end{remark}
\subsubsection{($c$)($wf$)-decomposition.}
Dually, the {\em (c)(wf)}-decomposition of $X\xra f Y$ leads us to consider
\newline\noindent
$
\xymatrix{
{X} \ar[r]\ar[d]|{(x,f(x))} &{ X\times X } \ar[r]\ar[l]\ar[d]|{(x_1,x_2)} & { X\times X\times X } \ar[r]\ar[l]\ar[d]|{(x_1,x_2,,x_3))} & {} \ar[l] \\
{(X\sqcup Y)} \ar[r]\ar[d]|{y} &{ (X\sqcup Y)\times (X\sqcup Y) } \ar[r]\ar[l]\ar[d]|{(y_1,y_2)} & { (X\sqcup Y)\times (X\sqcup Y)\times (X\sqcup Y) } \ar[r]\ar[l]\ar[d]|{(y_1,y_2,y_3)} & {} \ar[l] \\
{ Y} \ar[r] &{ Y\times Y } \ar[r]\ar[l] & { Y\times Y\times Y } \ar[r]\ar[l] & {} \ar[l] } \\
$
Intuitively, a (infinitesimal) neighbourhood of a point $y$ contains points $(x,1)$ whenever $f(x)=y$.
This motivates us to modify the topoic filter of the disjoint union topology on $X\sqcup Y$
by requiring its topoic subsets to satisfy also the following property:
$x\in P$ whenever $f(x)\in P$, $x\in X$.
That is, we define a new filter $\mathfrak F_{\text{c-wf}}^{n}$ on $(X\times Y)^n$
which consists of the subsets topoic wrt the disjoint union topology
which also satisfy the property that
$(z_1,..,x_i,...,z_n)\in P$ whenever $x_i\in X$ and $(z_1,..,f(x_i),...,z_n)\in P.$
Arguably, one might find an intuition according to which $(X\sqcup Y)_{\mathfrak F_{\text{c-wf}}}$ is similar to $Y_{\ttt}$.
\section{Open questions and directions for research}
\subsection{Research directions}
Our observations suggest the following broad questions and directions for research.
\subsubsection{Category theory implicit in elementary topology}
We'd like to think of our observations as {\em translations of ideas of
Bourbaki on general topology into a language of category theory appropriate to these ideas}, and
that these ideas (but not notation) are implicit in Bourbaki and reflect their
logic (or perhaps the ergologic in the sense of [Gromov]).
\begin{enonce}{Question}[Category theory and topological ideas and intuition]
\begin{itemize}
\item Translate more of Bourbaki and some intuitive topological arguments into
the language of category theory and diagram chasing.
\item Understand how this translation works and in what way it is a {\em translation}
rather than something new. Formulate what does it mean to say that these
category theoretic constructions are implicit in Bourbaki and find evidence
that indeed they are implicitly there.
\item More speculatively, find evidence that these category theoretic
diagram chasing arguments are implicitly present in the topological intuition of a student,
say by finding correlations between errors of intuition and errors of calculation.
\end{itemize}
\end{enonce}
But is this so and what does it actually mean?
The goal of our analysis is somewhat reminiscent of the goal of
\href{http://wilfridhodges.co.uk/arabic05.pdf}{[Hodges. Ibn Sina on analysis:
1. Proof search. Or: Abstract State Machines as a tool for history of logic]}
where he ``extract[s] from [the text of Ibn Sina's commentary on a couple of
paragraphs of Aristotle's Prior Analytics] all the essential ingredients of an
Abstract State Machine for [a proof search] algorithm''. We'd like to think
that we extract from the text of a couple of paragraphs of Bourbaki
all the essential ingredients of certain category theoretic constructions.
\subsubsection{Formalisation of topology.} Our translation is unsophisticated and is largely based
on textual coincidences and correlations between the text and allowed category theoretic manipulations.
Can these coincidences---and the translation---be found by a machine learning algorithm? A hope is that
category theoretic manipulations are restrictive enough so that a brute force search for correlations
between (long enough sequences of) allowed category theoretic manipulations and
the text of Bourbaki may produce meaningful results.
Designing such an algorithm would involve designing a derivation system for the category theoretic constructions used.
Our reformulations of certain notions of topology in terms of orthogonality (negation) are so concise (several bytes)
that they can be found, or rather listed, by a brute force search.
This might be a starting point in designing such an algorithm: first design
an algorithm which can work with the reformulations in terms of
iterated orthogonals (negations) of maps between finite spaces, %these reformulations,
find correlations between these orthogonals and text of Bourbaki,
and single out
the interesting notions obtained by iterated negation (orthogonals) from very simple morphism.
These notions should include quasi-compactness, denseness, connected etc.
\begin{enonce}{Question}[Category theory and topological ideas and intuition]
\begin{itemize}
\item Write a short program which extracts diagram chasing derivations from
texts on elementary topology, in the spirit of the ideology of ergosystems/ergostructures.
The texts might include \href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki, General Topology]} as well as some informal explanations.
In particular, it should be able to convert verbal definitions of properties defined by orthogonals
into the corresponding orthogonals.
\item Develop a formalisation of topology based on this translation.
\end{itemize}
\end{enonce}
\subsubsection{Tame topology and foundations of topology}
Does our point of view shed light on the tame topology of Grothendieck
and allows to develop a foundation of topology
``without false problems'' and ``wild phenomena'' ``at the very beginning''?
[Esquisse d'un Programme, translation,\S5,p.33]
Our approach does seem to avoid certain set-theoretic issues and constructions.
For example, ultrafilters do appear in our reformulation of compactness, but do so
only in a combinatorial disguise. Remark~\ref{ex:paths_as_Aut} in \S\ref{exp:limits} suggests a way to think about pathspaces
without real numbers. %(if it works).
\begin{enonce}{Question}[Tame topology and foundations of topology.]
\begin{itemize}
\item Develop elementary topology in terms of finite categories (viewed as
finite topological spaces) and labelled commutative diagrams, with an emphasis on
labels (properties) of morphisms defined by iterated orthogonals ($\rtt$-negation).
\item Develop topology in terms of finite categories, labelled commutative diagrams,
and simplicial filters. Develop a syntax to describe simplicial filters
as concise as the syntax of iterated $\rtt$-negation
of maps between finite spaces.
Does this
lead to tame topology of Grothendieck, i.e. a foundation of topology ``without false
problems'' and ``wild phenomena'' ``at the very beginning'' ?
\end{itemize}
\end{enonce}
Grothendieck suggests that the following needs to be done first:
\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}
%## Problem.
\begin{enonce}{Question} %\begin{enonce}{Question} %tex
Write a first year course introducing elementary topology
and category theory ideas at the same time,
based on the observations above
and the calculus to be developed.
Compactness would be explained with help of all the definitions above;
Tychonoff theorem follows immediately by a diagram chasing argument
from the fact that compactness is given by $\rtt$-negation (orthogonal);
$\AE\longrightarrow \EA$ definitions would give students some intuition.
As a first step, write an exposition aimed at students
of the separation axioms and Uryhson Lemma in terms
of the lifting properties.\footnote{
See \url{https://ncatlab.org/nlab/show/separation+axioms+in+terms+of+lifting+properties} for a list of reformulations of the separation axioms
in terms of orthogonals.}
\end{enonce}%\end{enonce} %tex
Note that the standard proof of Uryhson lemma can be represented as follows: iterate the lifting property defining normal ($T_4$) spaces
$$ \emptyset \longrightarrow {X} \,\rightthreetimes\, \{x{\swarrow}x'{\searrow}X{\swarrow}y'{\searrow}y\} \longrightarrow \{x{\swarrow}x'=X=y'{\searrow}y\}
$$
to prove
$$ \emptyset \longrightarrow X\rightthreetimes \{x \swarrow x_1 \searrow ... \swarrow x_n \searrow y \} \longrightarrow \{x \swarrow x_1 = ... = x_n \searrow y \}$$
Then pass to the infinite limit to construct a map
$ X \longrightarrow \mathbb{R}$.
\subsubsection{Homotopy and model category structure.}
Let $\FFilt$ be the category $\Filt$ of filters localised as follows:
we consider two morphisms equal iff they coincide on a big subset of the domain,
i.e.~$f,g:X\lra Y$ are considered equal as morphisms in $\FFilt$ iff
the subset $\{x: f(x)=g(x)\}$ is big in $X$.
\begin{enonce}{Question}[Homotopy theory and model category structure on $\sFilt$ or $\sFFilt$.]
Is there an interesting model category structure on $\sFilt$ or $\sFFilt$? %or a similar category?
Does it lead to interesting homotopy theory of uniform spaces?
In \S\ref{cwf-decompositions} we suggest examples of (cw)(f)- and (c)(wf)-decompositions.
Do the corresponding classes of acyclic cofibrations and fibrations
generate a model structure on $\sFilt$ or $\sFFilt$?
Does either category have interesting objects corresponding to quotients of topological spaces
by a group action?
\end{enonce}
\subsection{Metric spaces, uniform spaces and coarse spaces}
\subsubsection{Uniform structures}
%## Problem.
\begin{enonce}{Question}%\begin{enonce}{Question} %tex
Rewrite the theory of uniform structures and metric spaces
in terms of
the category $\sFilt$ of simplicial filters.
In particular,
\bi\item reformulate the Lebesgue's number lemma,
partition of unity, and the characterisation of paracompactness
by A.Stone mentioned by
\href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=6719&option_lang=eng}{[Alexandroff]}
(cf.~\S\ref{app:AEEA}).
\ei\end{enonce}%\end{enonce} %tex
\begin{enonce}{Question}[Arzela-Ascoli]\label{q: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\}) $, $\Ee(\NN_{cofinite}\cup_{\NN_{cofinite}}\{\infty\} )$,
$\ttt(\NN_{cofinite})$, $\ttt(X)$, $\mU(X)$, 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 various function spaces in terms of something like inner $Hom$ in $\sFilt$.
\eee
\end{enonce}
\subsubsection{Large scale geometry.}
The category of quasigeodesic metric spaces and large scale Lipschitz maps
embeds into another category $\sFmilt$ of simplicial filters, with maps of filters defined
differently: a $\Fmilt$-morphism of filters maps a small subset into a small subset.
%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 transformation $\mathcal X\lra \mathcal Y$ 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]|{#2} & {#4} \ar[d]|{#5} \\ {#3} \ar[r] \ar@{-->}[ur]^{}& {#6} }}
\begin{figure}
\begin{center}
\large
$ (a)\ \xymatrix{ A \ar[r]^{i} \ar@{->}[d]|f & X \ar[d]|g \\ B \ar[r]|-{j} \ar@{-->}[ur]|{{\tilde j}}& Y }$% \
%$\rrt ABXY$\ \ \ \
$(b)\ \rrt {\{\}} {} {\{\bullet\}} X {\therefore(surj)} Y $%\
$(c)\ \rrt {\{\bullet,\bullet\}} {} {\{\bullet\}} X {\therefore(inj)} Y $%\ \
$(d)\ \rrt X {\therefore(inj)} {Y} {\{x,y\}} {} {\{x=y\}}$\
\end{center}
\caption{\label{fig4}\normalsize
Lifting properties. Dots $\therefore$ indicate free variables and what property of these variables is being defined;
%, i.e.~a property of what is being defined and how is it to be labelled
in a diagram chasing calculation, ``$\therefore(surj)$" reads as:
given a (valid) diagram, add label $(surj)$ to the corresponding arrow.\newline
(a) The definition of a lifting property $f\rtt g$: for each $i:A\lra X$ and $j:B\lra Y$
making the square commutative, i.e.~$f\circ j=i\circ g$, there is a diagonal arrow $\tilde j:B\lra X$ making the total diagram
$A\xra f B\xra {\tilde j} X\xra g Y, A\xra i X, B\xra j Y$ commutative, i.e.~$f\circ \tilde j=i$ and $\tilde j\circ g=j$.
(b) $X\lra Y$ is surjective\newline
(c) $X\lra Y$ is injective; $X\lra Y$ is an epicmorphism if we forget %never use
that $\{\bullet\}$ denotes a singleton (rather than an arbitrary object
and thus $\{\bullet,\bullet\}\lra\{\bullet\}$ denotes an arbitrary morphism $Z\sqcup Z\xra{(id,id)} Z$)\newline
(d) $X\lra Y$ is injective, in the category of Sets; $\pi_0(X)\lra\pi_0(Y)$ is injective,
when the diagram is interpreted in the category
of topological spaces.
}
\end{figure}
So we rewrote these definitions without any words at all. Our benefits?
The usual little miracles happen:
The notation makes apparent a similarity of (*)${}_{\text{words}}$ and
(**)${}_{\text{words}}$: they are obtained, in the same purely formal way, from
the two of the simplest arrows (maps, morphisms) in the category of Sets. More is true:
it is also apparent that these two arrows are the simplest {\em counterexamples} to the properties,
and this suggests that we think of the lifting property as a category-theoretic (substitute for) negation.
Note also that a non-trivial (one which is not an non-isomorphism) morphism never has the lifting property
relative to itself, which fits with this interpretation.
Now that we have a formal notation and the little observation above,
we start to play around looking at simple arrows in various categories,
and also at not-so-simple arrows representing standard counterexamples.
You notice a few words from your first course on topology:
{\em %injective, surjective,
$(i)$ connected, $(ii)$ the separation axioms $T_0$ and $T_1$, $(iii)$ dense, $(iv)$ induced (pullback) topology}, and $(v)$ {\em Hausdorff %; surjective and injective on $\pi_0$
}
are, respectively,\footnote{The notation is self-explanatory; for the definition see \S\ref{app:top-notation}.}
%\newcommand*{\longhookrightarrow}{\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}}
%\def\lhra{\longhookrightarrow}
\bi
\item[(i):] $$ X\lra\{\bullet\}\rtt \{\bullet,\bullet\}\lra\{\bullet\}$$
\item[(ii):] $$\{\bullet \leftrightarrow \star\}\lra\{\bullet=\star\} \rtt X\lra\{\bullet\}$$ and
$$\{\bullet\ra \star\}\lra\{\bullet=\star\} \rtt X\lra\{\bullet\}$$
\item[(iii):] $$X\lra Y\rtt \{\bullet\}\lra\{\bullet\rightarrow\star\}$$
\item[(iv):] $$X\lra Y\rtt \{\bullet\ra\star\}\lra\{\bullet\}$$
\item[(v):] $$\{\bullet,\bullet'\}\xra{(inj)} X \rtt \{\bullet\leftarrow\star\rightarrow\bullet'\}\lra \{\bullet\}$$
\ei
%See the last two pages for illustrations how to read and draw on the blackboard
%these lifting properties in topology;
here
$$\{\bullet\ra\star\},\; \{\bullet\leftrightarrow\star\},\; \dots\ $$
denote finite preorders, or, equivalently,
finite categories with at most one arrow between any two objects, or finite topological spaces
on their elements or objects,
where a subset is closed iff it is downward closed (that is, together with each element, it contains all the smaller elements).
Thus $$\{\bullet\ra\star\},\; \{\bullet\leftrightarrow\star\} \;\mbox{ and }\; \{\bullet\leftarrow\star\rightarrow\bullet'\}$$
denote the connected spaces with only one open point $\bullet$, with no open points, and with two open points
$\bullet,\bullet'$ and a closed point $\star$.
Line (v) is to be interpreted somewhat differently: we consider {\em all}
the injective arrows of form $\{\bullet,\bullet'\}\lra X$.
We mentioned that the lifting property can be seen as a kind of negation. Confusingly, there are {\em two} negations, depending on whether the morphism appears on the left or right
side of the square, that are quite different: for example, both the pullback topology and the separation axiom $T_1$ are
negations of the same morphism, and the same goes for injectivity and injectivity on $\pi_0$ (see Figure~4(c,d)).
Now consider the standard example of something non-compact: the open covering
\[
\Bbb R= \bigcup\limits_{n\in\Bbb N} \{\,x\,:\, -nn$, $dist(``x_\infty\text{''},``x_n\text{''})=\frac1n$, as defined above.)
In functional analysis, a (partially defined!)
linear operator $f:X\lra Y$ between Banach spaces $X$ and $Y$
is {\em closed} iff for every convergent sequence $x_n\in X$, if $f(x_n)\xra[n\lra\infty]{}y$ in $Y$, % converges to some $y\in Y$ as $n$ goes to infinity,
then there is a $x\in X$ such that $f(x)=y$ and $x_n\xra[n\lra\infty]{}x$, i.e. % converges to $x$, i.e.
\[
\left\{``x_n\text{''}\,:\ n\in{\Bbb N}\right\}\lra\left\{``x_n\text{''}\,:\,n\in {\Bbb N}\right\}\cup\{``x_\infty\text{''}\} \rtt \text{Domain}(f)\lra Y
\]
A module $P$ over a commutative ring $R$ is {\em projective} iff for an arbitrary arrow $N\lra M$ in the category of $R$-modules it holds
$$ 0\lra R \rtt N\lra M \ \implies\ 0\lra P \rtt N\lra M
.$$
Dually, a module $I$ over a ring $R$ is {\em injective} iff for an arbitrary arrow $N\lra M$ in the category of $R$-modules it holds
$$ R\lra 0 \rtt N\lra M \ \implies\ N\lra M \rtt I\lra 0\\
.$$
\def\nto{\not\to}
\def\ZpZ{{\Bbb Z}/\!p{\Bbb Z}}
\subsubsection{Finite groups.} There are examples outside of topology; see Appendix~\ref{app:examples}.
Let us give some examples in group theory.
% cf.~Figure~2.
There is no non-trivial homomorphism from a group $F$ to $G$, write $F\nto G$, iff
$$0\lra F\,\rtt 0\lra G\text{ or equivalently }F\lra 0 \rtt G\lra 0.$$
A group $A$ is {\em Abelian} iff
\[ \left \,\lra\,\left \rtt\,\, A\lra 0
\]
where $\left \,\lra\,\left$ is the abelianisation morphism sending the free group into the Abelian free group on two generators;
a group $G$ is {\em perfect}, $G=[G,G]$, iff $G\nto A$ for any Abelian group $A$, i.e.
\[ \left \,\lra\,\left \rtt\,\, A\lra 0\ \implies\ G\lra 0 \rtt A\lra 0\]
in the category of finite or algebraic groups,
a group $H$ is {\em soluble} iff $G\nto H$ for each perfect group $G$,
i.e.
$$0\lra G\,\rtt 0\lra H\text{ or equivalently }C\lra 0 \rtt H\lra 0.$$
A prime number $p$ does not divide the number elements of a finite group $G$
iff $G$ has no element of order $p$, i.e. no element $x\in G$ such that $x^p=1_G$ yet $x^1\neq 1_G,...,x^{p-1}\neq 1_G$,
equivalently $\ZpZ\nto G$, i.e.
$$0\lra \ZpZ\,\rtt 0\lra G\text{ or equivalently }\ZpZ\lra 0 \rtt G\lra 0.$$
A finite group $G$ is a $p$-group, i.e. the number of its elements is a power of a prime number $p$, iff in the category of finite groups
$$0\lra \ZpZ \rtt 0\lra H \implies 0\lra H\rtt 0\lra G.$$
\subsection{Appendix. Iterated orthogonals: definitions and intuition.}
For a property (class) \ensuremath{C} of arrows (morphisms) in a category,
define {\em its left and right orthogonals}, which we also call
{\em left and right negation}:
$$ C^l := \{ \ensuremath{f} :\text{ for each }g \in C\ \ensuremath{f} \,\rightthreetimes\, \ensuremath{g} \} $$
$$ C^r := \{ \ensuremath{g} :\text{ for each }f \in C\ \ensuremath{f} \,\rightthreetimes\, \ensuremath{g} \} $$
$$ C^{lr}:=(C^l)^r, ... $$
here $f \,\rightthreetimes\, g$ reads ``$f$ has the left lifting property wrt $g$ '',
`` $f$ is (left) orthogonal to $g$ '',
i.e. for $f:A\longrightarrow B$, $g:X\longrightarrow Y$,
$f \,\rightthreetimes\, g$ iff for each $i:A\longrightarrow X$, $j:B\longrightarrow Y$ such that $ig=fj$ (``the square commutes''),
there is $j':B\longrightarrow X$ such that $fj'=i$ and $j'g=j$ (``there is a diagonal
making the diagram commute''), cf.~Fig.~\ref{fig:2a}.
The following observation is enough to reconstruct all the examples of
iterated orthogonals in this paper, with a bit of search and computation.
\begin{quote}
{\bf Observation.}\\
A number of elementary properties can be obtained by repeatedly passing
to the left or right orthogonal $C^l, C^r, C^{lr}, C^{ll}, C^{rl}, C^{rr},...$
starting from a simple class of morphisms, often
a single (counter)example to the property you define.
The counterexample is often implicit in the text of the definition of the property.
\end{quote}
A useful intuition is to think that the property of left-lifting against a
class \ensuremath{C} is a kind of negation of the property of being in \ensuremath{C}, and that
right-lifting is another kind of negation. Hence the classes obtained from C
by taking orthogonals an odd number of times, such as $C^l, C^r, C^{lrl}, C^{lll}$
etc., represent various kinds of negation of \ensuremath{C}, so $C^l, C^r, C^{lrl}, C^{lll}$ each
consists of morphisms which are far from having property $C$.
Taking the orthogonal of a class \ensuremath{C} is a simple way to define a class of
morphisms excluding non-isomorphisms from \ensuremath{C}, in a way which is useful in a
diagram chasing computation.
The class $C^l$ is always closed under retracts, pullbacks, (small) products
(whenever they exist in the category) and composition of morphisms, and
contains all isomorphisms of $C$. Meanwhile, $C^r$ is closed under retracts,
pushouts, (small) coproducts and transfinite composition (filtered colimits) of
morphisms (whenever they exist in the category), and also contains all
isomorphisms. Under some assumptions on existence of limits and colimits and ignoring
set-theoretic difficulties\footnote{For an example of a theorem along these lines see~\href{https://core.ac.uk/download/pdf/82479252.pdf}{[Bousfield, Constructions of factorization systems in categories, 5.1 Ex, 3.1 Thm]}. Note that he considers the {\em unique} lifting property, unlike us.},
each morphism $X\lra Y$ decomposes both as
$X\xra{(C)^l} \bullet \xra{(C)^{lr}} Y$ and $X\xra{(C)^{rl}} \bullet \xra{(C)^{r}} Y$.
For example, the notion of isomorphism can be obtained starting from the class
of all morphisms, or any single example of an isomorphism:
$$
(Isomorphisms) = (all\ morphisms)^l = (all\ morphisms)^r = (h)^{lr} = (h)^{rl}
$$
where \ensuremath{h } is an arbitrary isomorphism.
\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} }}
\begin{figure}
\begin{center}
$ (a)\ \xymatrix{ A \ar[r]|{i} \ar@{->}[d]|f & X \ar[d]|g \\ B \ar[r]|-{j} \ar@{-->}[ur]|{{\tilde j}}& Y }$% \
%$\rrt ABXY$\ \ \ \
$(b)\ \rrt {\{\}} {} {\{\bullet\}} X {\therefore(surj)} Y $%\
$(c)\ \rrt {\{\bullet,\bullet\}} {} {\{\bullet\}} X {\therefore(inj)} Y $%\ \
%$(d)\ \rrt X {\therefore(inj)} {Y} {\{x,y\}} {} {\{x=y\}}$\
\end{center}
\caption{\label{fig:2a}\normalsize
Lifting properties. %Dots $\therefore$ indicate free variables and what property of these variables is being defined;
%, i.e.~a property of what is being defined and how is it to be labelled
%in a diagram chasing calculation, ``$\therefore(surj)$'' reads as:
%given a (valid) diagram, add label $(surj)$ to the corresponding arrow.\newline
(a) The definition of a lifting property $f\rtt g$.
%: for each $i:A\lra X$ and $j:B\lra Y$
%making the square commutative, i.e.~$f\circ j=i\circ g$, there is a diagonal arrow $\tilde j:B\lra X$ making the total diagram
%$A\xra f B\xra {\tilde j} X\xra g Y, A\xra i X, B\xra j Y$ commutative, i.e.~$f\circ \tilde j=i$ and $\tilde j\circ g=j$.
(b) $X\lra Y$ is surjective %\newline
(c) $X\lra Y$ is injective
%; $X\lra Y$ is an epicmorphism if we forget %never use
%that $\{\bullet\}$ denotes a singleton (rather than an arbitrary object
%and thus $\{\bullet,\bullet\}\lra\{\bullet\}$ denotes an arbitrary morphism $Z\sqcup Z\xra{(id,id)} Z$)\newline
% (d) $X\lra Y$ is injective, in the category of Sets; $\pi_0(X)\lra\pi_0(Y)$ is injective if
% the diagram is interpreted in the category
%of topological spaces.
}
\end{figure}
\subsubsection{\label{app:rtt-examples}\label{app:examples}Examples of iterated orthogonals.}
Here give a list of examples of well-known properties which can be defined by
iterated orthogonals starting from a simple class of morphisms.
\begin{itemize}
\item[ (i)] $(\emptyset\longrightarrow \{*\})^r$, $(0\longrightarrow R)^r$, and $\{0\longrightarrow \ZZ\}^r$ are the classes of surjections in
in the categories of Sets, $R$-modules, and Groups, resp.,
(where $\{*\}$ is the one-element set, and in the category of (not necessarily abelian) groups, $0$ denotes the trivial group)
\item[ (ii)] $(\{\star,\bullet\}\longrightarrow \{*\})^l=(\{\star,\bullet\}\longrightarrow \{*\})^r$, $(R\longrightarrow 0)^r$, $\{\ZZ\longrightarrow 0\}^r$ are the classes of
injections in the categories of Sets, $R$-modules, and Groups, resp
\item[ (iii)] in the category of $R$-modules, \\
\ \ a module \ensuremath{P} is projective iff $0\longrightarrow P$ is in $(0\longrightarrow R)^{rl}$\\
\ \ a module $I$ is injective iff $I\longrightarrow 0$ is in $(R\longrightarrow 0)^{rr}$
\item[ (iv)] in the category of Groups, \begin{itemize}
\item[] a finite group \ensuremath{H} is nilpotent iff $H\longrightarrow H\times H$ is in $\{\, 0\longrightarrow \ensuremath{G} : G\text{ arbitrary} \}^{lr}$
\item[] a finite group \ensuremath{H} is solvable iff $0\longrightarrow H$ is in $\{\, 0\longrightarrow A : A\text{ abelian }\}^{lr}= \{\, [G,G]\longrightarrow \ensuremath{G} : G\text{ arbitrary }\}^{lr}$
%\item[] a finite group \ensuremath{H} is of order prime to $p$ iff $H\longrightarrow 0$ is in $\{\ZZ/p\ZZ\longrightarrow 0\}^r$
\item[] a finite group \ensuremath{H} is a $p$-group iff $H\longrightarrow 0$ is in $\{\ZZ/p\ZZ\longrightarrow 0\}^{rr}$
%%\item[] a group \ensuremath{H} is torsion-free iff $0\longrightarrow H$ is in $\{ n\ZZ\longrightarrow \ZZ: n>0 \}^r$
\item[] a group \ensuremath{F} is free iff $0\longrightarrow F$ is in $\{0\longrightarrow \ZZ\}^{rl}$
%%\item[] a homomorphism $f$ is split iff $f \in \{\, 0\longrightarrow \ensuremath{G} : G\text{ arbitrary} \}^r$
%+ a subgroup A is pure in a group $H$ iff $A\longrightarrow H$ is in $\{ n\ZZ\longrightarrow \ZZ : n>0 \}^r$
\end{itemize}
\item[ (v)] in the category of metric spaces and uniformly continuous maps,\\
a metric space \ensuremath{X} is complete iff $\{1/n\}_n\longrightarrow \{1/n\}_n\cup \{0\} \,\rightthreetimes\, X\longrightarrow \{0\}$
where the metric on $\{1/n\}_n$ and $\{1/n\}_n\cup \{0\}$ is induced from the
real line\\
a subset $A \subset X$ is closed iff $\{1/n\}_n\longrightarrow \{1/n\}_n\cup \{0\} \,\rightthreetimes\, A\longrightarrow X$
\item[ (vi)] in the category of topological spaces,\\
for a connected topological space $X$, each function on \ensuremath{X} is bounded
iff %\emptyset\longrightarrow \ensuremath{X} \,\rightthreetimes\, |_|_n (-n,n) \longrightarrow R
$$ \emptyset\longrightarrow \ensuremath{X} \,\rightthreetimes\, \cup_n (-n,n) \longrightarrow \RR$$
%\newpage
\item[ (vii)] in the category of topological spaces (see notation defined below),
\begin{itemize}
\item[] a space $X$ is path-connected iff $\{0,1\} \longrightarrow [0,1] \,\rightthreetimes\, \ensuremath{X} \longrightarrow \{*\}$
\item[] a space $X$ is path-connected iff for each Hausdorff compact space $K$ and each injective map $\{x,y\} \hookrightarrow K$ it holds
$\{x,y\} \hookrightarrow \ensuremath{K} \,\rightthreetimes\, \ensuremath{X} \longrightarrow \{*\}$
\end{itemize}
\end{itemize}
{\bf
Proof.
} In (iv), we use that a finite group $H$ is nilpotent iff the diagonal $\{
(h,h) : \ensuremath{h } \in \ensuremath{H} \}$ is subnormal in $H\times H$,
cf.~\href{http://groupprops.subwiki.org/wiki/Nilpotent group}{[Nilp]}.
\qed
\subsection{A concise notation for certain properties in elementary point-set topology}
We introduce a concise, and in a sense intuitive, notation (syntax) able to express
a number of properties in elementary point-set topology. It is appropriate
for properties defined as iterated orthogonals (negation) starting from maps of finite topological spaces.\footnote
{I thank Urs Schreiber for help with the exposition in this subsection.}
For example, surjective, injective, connected, totally disconnected, and dense %and being a proper map
are expressed as
$\{\{\}\lra \{a\}\}^r$, $\{\{x,y\}\lra \{x=y\}\}^r$,
$\{\{x,y\}\lra \{x=y\}\}^l$ or $\{\{\}\lra \{a\}\}^{rll}$,
$\{\{\}\lra \{a\}\}^{rllr}$,
$\{\{x\}\lra \{x\swarrow y\}\}^r$.
\subsubsection{\label{app:top-notation}Notation for maps between finite topological spaces.
%\footnote{I thank Urs Schreiber for help with the exposition in this subsection.}
}
A {\em topological space} comes with a {\em specialisation preorder} on its points: for
points $x,y \in X$, $x \leq y$ iff $y \in cl x$ ($y$ is in the {\em topological closure} of $x$).
The resulting {\em preordered set} may be regarded as a {\em category} whose
{\em objects} are the points of ${X}$ and where there is a unique {\em morphism} $x{\searrow}y$ iff $y \in cl x$.
For a {\em finite topological space} $X$, the specialisation preorder or
equivalently the corresponding category uniquely determines the space: a {\em
subset} of ${X}$ is {\em closed} iff it is
{\em downward closed}, or equivalently,
is a full subcategory such that there are no morphisms going outside the subcategory.
The monotone maps (i.e. {\em functors}) are the {\em continuous maps} for this topology.
We denote a finite topological space by a list of the arrows (morphisms) in
the corresponding category; '$\leftrightarrow $' denotes an {\em isomorphism} and '$=$' denotes the {\em identity morphism}. An arrow between two such lists denotes a {\em continuous map} (a functor) which sends each point to the correspondingly labelled point, but possibly turning some morphisms into identity morphisms, thus gluing some points.
With this notation, we may display continuous functions for instance between the {\em discrete space} on two points, the {\em Sierpinski space}, the {\em antidiscrete space} and the {\em point space} as follows (where each point is understood to be mapped to the point of the same name in the next space):
$$
\begin{array}{ccccccc}
\{a,b\}
&\longrightarrow&
\{a{\searrow}b\}
&\longrightarrow&
\{a\leftrightarrow b\}
&\longrightarrow&
\{a=b\}
\\
\text{(discrete space)}
&\longrightarrow&
\text{(Sierpinski space)}
&\longrightarrow&
\text{(antidiscrete space)}
&\longrightarrow&
\text{(single point)}
\end{array}
$$
In $A \longrightarrow B$, each object and each morphism in $A$ necessarily appears in ${B}$ as well; we avoid listing
the same object or morphism twice. Thus
both
$$
\{a\} \longrightarrow \{a,b\}
\phantom{AAA} \text{ and } \phantom{AAA}
\{a\} \longrightarrow \{b\}
$$
denote the same map from a single point to the discrete space with two points.
Both
$$\{a{\swarrow}U{\searrow}x{\swarrow}V{\searrow}b\}\longrightarrow \{a{\swarrow}U=x=V{\searrow}b\}\text{ and }\{a{\swarrow}U{\searrow}x{\swarrow}V{\searrow}b\}\longrightarrow \{U=x=V\}$$
denote the morphism gluing points $U,x,V$.
In $\{a{\searrow}b\}$, the point $a$ is open and point ${b}$ is closed. We
denote points by $a,b,c,..,U,V,...,0,1..$ to make notation reflect the intended meaning,
e.g.~$X\lra \{U\searrow U'\}$ reminds us that the preimage of $U$ determines an open subset of $X$,
$\{x,y\}\lra X$ reminds us that the map determines points $x,y\in X$, and $\{o\searrow c\}$ reminds that $o$ is open and $c$ is closed.
Each continuous map $A\lra B$ between finite spaces may be represented in this way; in the first list
list relations between elements of $A$, and in the second list put relations between their images.
However, note that this notation does not allow to represent {\em endomorphisms $A\lra A$}.
We think of this limitation
as a feature and not a bug: in a diagram chasing computation,
endomorphisms under transitive closure lead to infinite cycles,
and thus our notation has better chance to define a computable fragment of topology.
\subsubsection{\label{app:rtt-top}Examples of iterated orthogonals obtained from maps between finite topological spaces.}
Here give a list of examples of well-known properties which can be defined by
iterated orthogonals starting from maps between finite topological spaces, often with less than 5 elements.
In the category of topological spaces (see notation defined below),
\begin{itemize}
\item a Hausdorff space $K$ is compact iff $K\longrightarrow \{*\}$ is in $((\{o\}\longrightarrow \{o{\small\searrow}c\})^r_{<5})^{lr}$
\item a Hausdorff space $K$ is compact iff $K\longrightarrow \{*\}$ is in $$
\{\, \{a\leftrightarrow b\}\longrightarrow \{a=b\},\, \{o{\small\searrow}c\}\longrightarrow \{o=c\},\,
\{c\}\longrightarrow \{o{\small\searrow}c\},\,\{a{\small\swarrow}o{\small\searrow}b\}\longrightarrow \{a=o=b\}\,\,\}^{lr}$$
\item a space $D$ is discrete iff $ \emptyset \longrightarrow D$ is in $ (\emptyset\longrightarrow \{*\})^{rl} $
\item a space $D$ is antidiscrete iff $ \ensuremath{D} \longrightarrow \{*\} $ is in %\begin{center}
$(\{a,b\}\longrightarrow \{a=b\})^{rr}= (\{a\leftrightarrow b\}\longrightarrow \{a=b\})^{lr} $ %\end{center}
\item a space $K$ is connected or empty iff $K\longrightarrow \{*\}$ is in $(\{a,b\}\longrightarrow \{a=b\})^l $
\item a space $K$ is totally disconnected and non-empty iff $K\longrightarrow \{*\}$ is in $(\{a,b\}\longrightarrow \{a=b\})^{lr} $
\item a space $K$ is connected and non-empty
iff
for some arrow $\{*\}\longrightarrow K$\\
$\text{ \ \ \ \ \ } \{*\}\longrightarrow K$ is in
$ (\emptyset\longrightarrow \{*\})^{rll} = (\{a\}\longrightarrow \{a,b\})^l$
\item a space $K$ is non-empty iff $K\longrightarrow \{*\}$ is in $ (\emptyset\longrightarrow \{*\})^l$
\item a space $K$ is empty iff $K \longrightarrow \{*\}$ is in $ (\emptyset\longrightarrow \{*\})^{ll}$
\item a space $K$ is $T_0$ iff $K \longrightarrow \{*\}$ is in $ (\{a\leftrightarrow b\}\longrightarrow \{a=b\})^r$
\item a space $K$ is $T_1$ iff $K \longrightarrow \{*\}$ is in $ (\{a{\small\searrow}b\}\longrightarrow \{a=b\})^r$
\item a space $X$ is Hausdorff iff for each injective map $\{x,y\} \hookrightarrow X$
it holds $\{x,y\} \hookrightarrow \ensuremath{X} \,\rightthreetimes\, \{ \ensuremath{x} {\small\searrow} \ensuremath{o} {\small\swarrow} \ensuremath{y} \} \longrightarrow \{ x=o=y \}$
\item a non-empty space $X$ is regular (T3) iff for each arrow $ \{x\} \longrightarrow X$ it holds
$ \{x\} \longrightarrow \ensuremath{X} \,\rightthreetimes\, \{x{\small\searrow}X{\small\swarrow}U{\small\searrow}F\} \longrightarrow \{x=X=U{\small\searrow}F\}$
\item a space $X$ is normal (T4) iff $\emptyset \longrightarrow \ensuremath{X} \,\rightthreetimes\, \{a{\small\swarrow}U{\small\searrow}x{\small\swarrow}V{\small\searrow}b\}\longrightarrow \{a{\small\swarrow}U=x=V{\small\searrow}b\}$
\item a space $X$ is completely normal iff $\emptyset\longrightarrow \ensuremath{X} \,\rightthreetimes\, [0,1]\longrightarrow \{0{\small\swarrow}x{\small\searrow}1\}$
where the map $[0,1]\longrightarrow \{0{\small\swarrow}x{\small\searrow}1\}$ sends $0$ to $0$, $1$ to $1$, and the rest $(0,1)$ to $x$
\item a space $X$ is path-connected iff $\{0,1\} \longrightarrow [0,1] \,\rightthreetimes\, \ensuremath{X} \longrightarrow \{*\}$
\item a space $X$ is path-connected iff for each Hausdorff compact space $K$ and each injective map $\{x,y\} \hookrightarrow K$ it holds
$\{x,y\} \hookrightarrow \ensuremath{K} \,\rightthreetimes\, \ensuremath{X} \longrightarrow \{*\}$
\item a non-empty space $X$ is regular (T3) iff for each arrow $ \{x\} \longrightarrow X$ it holds
$ \{x\} \longrightarrow \ensuremath{X} \,\rightthreetimes\, \{x{\small\searrow}X{\small\swarrow}U{\small\searrow}F\} \longrightarrow \{x=X=U{\small\searrow}F\}$
\item a space $X$ is normal (T4) iff $\emptyset \longrightarrow \ensuremath{X} \,\rightthreetimes\, \{a{\small\swarrow}U{\small\searrow}x{\small\swarrow}V{\small\searrow}b\}\longrightarrow \{a{\small\swarrow}U=x=V{\small\searrow}b\}$
\item a space $X$ is completely normal iff $\emptyset\longrightarrow \ensuremath{X} \,\rightthreetimes\, [0,1]\longrightarrow \{0{\small\swarrow}x{\small\searrow}1\}$
where the map $[0,1]\longrightarrow \{0{\small\swarrow}x{\small\searrow}1\}$ sends $0$ to $0$, $1$ to $1$, and the rest $(0,1)$ to $x$
\item a space $X$ is path-connected iff $\{0,1\} \longrightarrow [0,1] \,\rightthreetimes\, \ensuremath{X} \longrightarrow \{*\}$
\item a space $X$ is path-connected iff for each Hausdorff compact space $K$ and each injective map $\{x,y\} \hookrightarrow K$ it holds
$\{x,y\} \hookrightarrow \ensuremath{K} \,\rightthreetimes\, \ensuremath{X} \longrightarrow \{*\}$
\item $(\emptyset\longrightarrow \{*\})^r$ is the class of surjections
\item $(\emptyset\longrightarrow \{*\})^r$ is the class of maps $A\lra B$ where $A\neq \emptyset$ or $A=B$
\item $(\emptyset\longrightarrow \{*\})^{rr}$ is the class of subsets, i.e. injective maps $A\hookrightarrow B$ where the topology on $A$ is induced from $B$
\item $(\emptyset\longrightarrow \{*\})^{lr}$ is the class of maps $\emptyset\longrightarrow B$, $B$ arbitrary
\item $(\emptyset\longrightarrow \{*\})^{lrl}$ is the class of maps $A\longrightarrow B$ which admit a section
\item $(\emptyset\longrightarrow \{*\})^l$ consists of maps $f:A\longrightarrow B$ such that either $A\neq \emptyset$ or $A=B=\emptyset$
\item $(\emptyset\longrightarrow \{*\})^{rl}$ is the class of maps of form $A\longrightarrow A\sqcup D$ where $D$ is discrete
\item $\{\bullet\}\lra A$ is in $(\emptyset\longrightarrow \{*\})^{rll}$ iff $A$ is connected
\item
$Y$ is totally disconnected iff $\{\bullet\}\xra y Y$ is in $(\emptyset\longrightarrow \{*\})^{rllr}$ for each map $\{\bullet\}\xra y Y$ (or,
in other words, each point $y\in Y$).
\item $(\{b\}\longrightarrow \{a{\small\searrow}b\})^l$ is the class of maps with dense image
\item $(\{b\}\longrightarrow \{a{\small\searrow}b\})^{lr}$ is the class of closed subsets $A \subset X$, $A$ a closed subset of $X$
%+ $(\emptyset\longrightarrow \{*\})^l$ is the class of maps $A\longrightarrow B$ such that $A=B=\emptyset$ or $A\neq\emptyset$, $B$ arbitrary
%+ $(\emptyset\longrightarrow \{*\})^{ll}$ is the class of maps $A\longrightarrow B$ such that either $A=\emptyset$ or the map is an isomorphism
%+ $(\emptyset\longrightarrow \{*\})^r$ is the class of surjections
%+ $(\emptyset\longrightarrow \{a\})^{rl}$ is the class of maps of form $A\longrightarrow A\cup D$, $D$ is discrete
\item $( \{a{\small\searrow}b\}\lra\{a=b\})^l$ is the class of injections
%\item $( \{a{\small\searrow}b\}\lra\{a=b\}$)^{ll}$
\item $((\{a\}\longrightarrow \{a{\small\searrow}b\})^r_{<5})^{lr}$ is roughly the class of proper maps
(see below).
\end{itemize}
{\bf
Proof.
}
Items related to compactness and proper maps are discussed in ??.
Other items require a simple if tedious verification. \qed
\subsection{Separation axioms as orthogonals.}
See \url{https://ncatlab.org/nlab/show/separation+axioms+in+terms+of+lifting+properties} for a list of reformulations of the separation axioms.
%%%---
%%%--- \subsubsection{Separation conditions in terms of arrows}
%%%---
%%%---
%%%--- Fix two sets ({\em subsets}) $F$ and $G$ of $S$.
%%%---
%%%--- \begin{itemize}
%%%--- \item The sets $F$ and $G$ are {\em disjoint sets|disjoint} if their {\em intersection} is {\em empty set|empty}:
%%%--- $$
%%%--- F \cap G = \emptyset
%%%--- $$
%%%---
%%%--- In terms of arrows, the following map is well-defined: $S_{F,G}: S \longrightarrow \{F\leftrightarrow G \leftrightarrow \bullet \}$ such that $S_{F,G}(x)=F$ for $x \in F$, $S_{F,G}(x)=G$ for $x \in G$, and $S_{F,G}(x)=\bullet$ for $x \notin F\cup G$.
%%%---
%%%--- \item They are {\em topologically disjoint} if there exists a {\em neighbourhood} of one set that is disjoint from the other set:
%%%--- $$
%%%--- (\exists\; U \stackrel{\circ}\supseteq F,\; U \cap G = \emptyset)
%%%--- \;\vee\;
%%%--- (\exists\; V \stackrel{\circ}\supseteq G,\; F \cap V = \emptyse)
%%%--- \,.
%%%--- $$
%%%---
%%%--- In terms of arrows, $S_{F,G}: S \longrightarrow \{F\leftrightarrow \bullet \leftrightarrow G \}$ factors either as
%%%---
%%%--- $$ S_{F,G}: S \longrightarrow \{F \leftrightarrow U \searrow \bullet \leftrightarrow G\} \longrightarrow \{F\leftrightarrow U =\bullet \leftrightarrow G \}$$
%%%---
%%%--- or
%%%---
%%%--- $$
%%%--- S_{F,G}
%%%--- \colon
%%%--- S \longrightarrow \{F \leftrightarrow \bullet \swarrow V \leftrightarrow G \}
%%%--- \longrightarrow
%%%--- \{F \leftrightarrow \bullet = V \leftrightarrow G \}
%%%--- $$
%%%---
%%%--- where $U$, $V$ maps to $\bullet$.
%%%--- Notice that topologically disjoint sets must be disjoint.
%%%---
%%%--- \item They are{\em separated}if each set has a neighbourhood that is disjoint from the other set:
%%%---
%%%--- $$
%%%--- (\exists\; U \stackrel{\circ}\supseteq F,\; U \cap G = \empty)
%%%--- \;\wedge\;
%%%--- (\exists\; V \stackrel{\circ}\supseteq G,\; F \cap V = \empty)
%%%--- \;\;\equiv\;\;
%%%--- \exists\; U \stackrel{\circ}\supseteq F,\; \exists\; V \stackrel{\circ}\supseteq G,\; U \cap G = \empty \;\wedge\; F \cap V = \empty
%%%--- \,.
%%%--- $$
%%%---
%%%--- In terms of arrows, $S_{F,G}: S \longrightarrow \{F\leftrightarrow \bullet \leftrightarrow G \}$ factors both as
%%%---
%%%--- $$
%%%--- S_{F,G}
%%%--- \colon
%%%--- S \longrightarrow \{F \leftrightarrow U \searrow \bullet \leftrightarrow G\} \longrightarrow \{F\leftrightarrow U \stackrel{\circ}=\bullet \leftrightarrow G \}
%%%--- $$
%%%---
%%%--- and as
%%%---
%%%--- $$
%%%--- S_{F,G}
%%%--- \colon
%%%--- S \longrightarrow \{F \leftrightarrow \bullet \swarrow V \leftrightarrow G \} \longrightarrow \{F \leftrightarrow \bullet = V \leftrightarrow G \}
%%%--- $$
%%%---
%%%--- where $ U $, $V $ maps to $\bullet$.
%%%--- Notice that separated sets must be topologically disjoint.
%%%---
%%%--- \item They are {\em separated by neighbourhoods} if they have disjoint neighbourhoods:
%%%--- $$ \exists\; U \stackrel{\circ}\supseteq F,\; \exists\; V \stackrel{\circ}\supseteq G,\; U \cap V = \empty .$$
%%%--- The arrow $S_{F,G}: S \longrightarrow \{F\leftrightarrow \bullet \leftrightarrow G \}$ factors as
%%%--- $S \longrightarrow \{F \leftrightarrow U \searrow \bullet \swarrow V \leftrightarrow G\}\longrightarrow \{F \leftrightarrow U = \bullet = V \leftrightarrow G\}$
%%%--- Notice that sets separated by neighbourhoods must be separated.
%%%--- \item They are{\em separated by closed neighbourhoods}if they have disjoint closed neighbourhoods:
%%%--- $$ \exists\; U \stackrel{\circ}\supseteq F,\; \exists\; V \stackrel{\circ}\supseteq G,\; Cl(U) \cap Cl(V) = \empty .$$
%%%--- The arrow $S_{F,G}: S \longrightarrow \{F\leftrightarrow \bullet \leftrightarrow G \}$ factors as
%%%--- $$ S \longrightarrow \{ F \leftrightarrow U \searrow U' \swarrow \bullet \searrow V' \searrow V \leftrightarrow G \}
%%%--- \longrightarrow \{ F \leftrightarrow U = U' = \bullet = V' = V \leftrightarrow G \} $$
%%%--- Notice that sets separated by closed neighbourhoods must be separated by neighbourhoods.
%%%--- \item They are{\em separated by a function}if there exists a continuous {\em real number|real}-valued {\em function} on the space that maps $F$ to $0$ and $G$ to $1$:
%%%--- $$ \exists\; f: S \to \mathbf{R},\; F \subseteq f^{-1} (\{0\}) \;\wedge\; G \subseteq f^{-1} (\{1\}) .$$
%%%--- The arrow $S_{F,G}: S \longrightarrow \{F\leftrightarrow \bullet \leftrightarrow G \}$ factors as
%%%--- $$ S \longrightarrow \{0'\} \cup [0,1] \cup \{1'\} \longrightarrow \{ 0'=F \leftrightarrow \bullet \leftrightarrow 1'=G\}
%%%--- $$
%%%--- where points $0',0$ and $1,1'$ are topologically indistinguishable,
%%%--- and $0'$ maps to $F$, and $1'$ maps to $G$, and $[0,1]$ maps to $\bullet$.
%%%--- Notice that sets separated by a function must be separated by closed neighbourhoods (the preimages of $[-\varepsilon, \varepsilon]$ and $[1-\varepsilon, 1+\varepsilon]$).
%%%--- \item Finally, they are{\em precisely separated by a function} if there exists a continuous real-valued
%%%--- function on the space that maps precisely $F$ to $0$ and $G$ to $1$:
%%%--- $$ \exists\; f: S \to \mathbf{R},\; F = f^{-1} (\{0\}) \;\wedge\; G = f^{-1} (\{1\}) .$$
%%%--- The arrow $S_{F,G}: S \longrightarrow \{F\leftrightarrow \bullet \leftrightarrow G \}$ factors as
%%%--- $$ S \longrightarrow [0,1] \longrightarrow \{ 0=F \leftrightarrow \bullet\leftrightarrow 1=G\}
%%%--- $$
%%%--- where
%%%--- $0'$ maps to $F$, and $1'$ maps to $G$, and $(0,1)$ maps to $\bullet$.
%%%--- Notice that sets separated by a function must be separated by closed neighbourhoods (the preimages of $[-\varepsilon, \varepsilon]$ and $[1-\varepsilon, 1+\varepsilon]$).
%%%--- Notice that sets precisely separated by a function must be separated by a function.
%%%--- \end{itemize}
%%%---
%%%--- Often $F$ and $G$ will be points (identified with their {\em singleton} subsets); in that case, one usually says {\em distinct}
%%%--- in place of {\em disjoint}.
%%%---
%%%--- Often $F$ or $G$ will be closed sets; notice that disjoint closed sets are automatically separated, while a closed set and a point, if disjoint, are automatically topologically disjoint.
%%%---
%%%---
%%%---
%%%--- \subsubsection{Separation axioms as lifting properties}
%%%---
%%%--- In all of the following definitions, ${X}$ is a topological space.
%%%---
%%%--- \begin{itemize}
%%%--- \item $ {X}$ is T0, or Kolmogorov, if any two distinct points in ${X}$ are topologically
%%%--- distinguishable. (It will be a common theme among the separation axioms to have
%%%--- one version of an axiom that requires T0 and one version that doesn't.)
%%%--- As a formula, this is expressed as
%%%--- $$ \{x\leftrightarrow y\} \longrightarrow \{x=y\} \,\rightthreetimes\, {X} \longrightarrow \{*\}$$
%%%---
%%%--- \item ${X}$ is R0, or symmetric, if any two topologically distinguishable points in $X$
%%%--- are separated, i.e.
%%%--- $$ \{x{\searrow}y\} \longrightarrow \{x\leftrightarrow y\} \,\rightthreetimes\, {X} \longrightarrow \{*\} $$
%%%---
%%%--- \item $X$ is T1, or accessible or Frechet, if any two distinct points in $X$ are
%%%--- separated, i.e.
%%%--- $$ \{x{\searrow}y\} \longrightarrow \{x=y\} \,\rightthreetimes\, {X} \longrightarrow \{*\} $$
%%%--- Thus, $X$ is T1 if and only if it is both T0 and R0. (Although you may
%%%--- say such things as "T1 space", "Frechet topology", and "Suppose that the
%%%--- topological space $X$ is Frechet", avoid saying "Frechet space" in this
%%%--- context, since there is another entirely different notion of Frechet space in
%%%--- functional analysis.)
%%%---
%%%--- \item $X$ is R1, or preregular, if any two topologically distinguishable points in
%%%--- X are separated by neighbourhoods. Every R1 space is also R0.
%%%---
%%%--- \item $X$ is weak Hausdorff, if the image of every continuous map from a compact
%%%--- Hausdorff space into $X$ is closed. All weak Hausdorff spaces are T1, and all
%%%--- Hausdorff spaces are weak Hausdorff.
%%%---
%%%--- \item $X$ is Hausdorff, or T2 or separated, if any two distinct points in $X$ are
%%%--- separated by neighbourhoods, i.e.
%%%--- $$ \{x,y\} \hookrightarrow {X} \,\rightthreetimes\, \{x{\searrow}X{\swarrow}y\} \longrightarrow \{x=X=y\} $$
%%%--- Thus, $X$ is Hausdorff if and only if it is both T0
%%%--- and R1. Every Hausdorff space is also T1.
%%%---
%%%--- \item $X$ is $T2\frac{1}{2}$, or Urysohn, if any two distinct points in $X$ are separated by
%%%--- closed neighbourhoods, i.e.
%%%--- $$
%%%--- \{x,y\} \hookrightarrow {X} \,\rightthreetimes\, \{x{\searrow}x'{\swarrow}X{\searrow}y'{\swarrow}y\} \longrightarrow \{x=x'=X=y'=y\}
%%%--- $$
%%%--- Every $T2\frac{1}{2}$ space is also Hausdorff.
%%%---
%%%--- \item $X$ is completely Hausdorff, or completely T2, if any two distinct points in
%%%--- X are separated by a continuous function, i.e.
%%%--- $$ \{x,y\} \hookrightarrow {X} \,\rightthreetimes\, [0,1]\longrightarrow \{*\}
%%%--- $$
%%%--- where $\{x,y\} \hookrightarrow X$ runs through all injective maps from the discrete two
%%%--- point space $\{x,y\}$.
%%%---
%%%--- Every completely Hausdorff space is
%%%--- also $ T2\frac{1}{2} $.
%%%---
%%%--- \item $X$ is regular if, given any point ${x}$ and closed subset $F$ in $X$ such that ${x}$ does
%%%--- not belong to $F$, they are separated by neighbourhoods, i.e.
%%%--- $$ \{x\} \longrightarrow {X} \,\rightthreetimes\, \{x{\searrow}X{\swarrow}U{\searrow}F\} \longrightarrow \{x=X=U{\searrow}F\}
%%%--- $$
%%%--- (In fact, in a regular
%%%--- space, any such ${x}$ and ${F}$ will also be separated by closed neighbourhoods.) Every
%%%--- regular space is also R1.
%%%---
%%%--- \item $X$ is regular Hausdorff, or T3, if it is both T0 and regular.[1] Every
%%%--- regular Hausdorff space is also $T2\frac{1}{2}$.
%%%---
%%%--- \item $X$ is completely regular if, given any point ${x}$ and closed set $F$ in $X$ such
%%%--- that ${x}$ does not belong to $F$, they are separated by a continuous function, i.e.
%%%--- $$
%%%--- \{x\} \longrightarrow {X} \,\rightthreetimes\, [0,1] \cup \{F\} \longrightarrow \{x{\searrow}F\}
%%%--- $$
%%%--- where points $F$ and $1$ are topologically indistinguishable, $[0,1]$ goes to $x$, and $F$ goes to $F$.
%%%---
%%%--- Every
%%%--- completely regular space is also regular.
%%%---
%%%--- \item $X$ is Tychonoff, or T3$\frac{1}{2}$, completely T3, or completely regular Hausdorff, if
%%%--- it is both T0 and completely regular.[2] Every Tychonoff space is both regular
%%%--- Hausdorff and completely Hausdorff.
%%%---
%%%--- \item $X$ is normal if any two disjoint closed subsets of $X$ are separated by
%%%--- neighbourhoods, i.e.
%%%--- $$ \emptyset \longrightarrow {X} \,\rightthreetimes\, \{x{\swarrow}x'{\searrow}X{\swarrow}y'{\searrow}y\} \longrightarrow \{x{\swarrow}x'=X=y'{\searrow}y\}
%%%--- $$
%%%--- In fact, by Urysohn lemma a space is normal if and only if any two disjoint
%%%--- closed sets can be separated by a continuous function, i.e.
%%%--- $$ \emptyset \longrightarrow {X} \,\rightthreetimes\, \{0'\} \cup [0,1] \cup \{1'\} \longrightarrow \{0=0'{\searrow}x{\swarrow}1=1'\} $$
%%%--- where points
%%%--- $0',0$ and $1,1'$ are topologically indistinguishable,
%%%--- $[0,1]$ goes to $x$, and both $0,0'$ map to point $0=0'$, and both $1,1'$ map to point
%%%--- $1=1'$.
%%%---
%%%---
%%%---
%%%--- \item $X$ is normal Hausdorff, or T4, if it is both T1 and normal. Every normal
%%%--- Hausdorff space is both Tychonoff and normal regular.
%%%---
%%%--- \item $X$ is completely normal if any two separated sets $A$ and $B$ are separated by
%%%--- neighbourhoods $U\supset A$ and $V\supset B$
%%%--- such that $U$ and $V$ do not intersect, i.e.
%%%--- $$\emptyset \longrightarrow {X} \,\rightthreetimes\, \{X{\swarrow}A\leftrightarrow U{\searrow}U'{\swarrow}W{\searrow}V'{\swarrow}V\leftrightarrow B{\searrow}X\} \longrightarrow \{U=U',V'=V\}$$
%%%--- Every completely normal space is also normal.
%%%---
%%%---
%%%--- \item $X$ is perfectly normal if any two disjoint closed sets are precisely
%%%--- separated by a continuous function, i.e.
%%%--- $$
%%%--- \emptyset\longrightarrow {X} \,\rightthreetimes\, [0,1]\longrightarrow \{0{\swarrow}X{\searrow}1\}
%%%--- $$
%%%--- where $(0,1)$ goes to the open point $X$, and $0$ goes to $0$, and $1$ goes to $1$.
%%%---
%%%--- Every perfectly normal space is also
%%%--- completely normal.
%%%--- \end{itemize}
%%%---
%%%--- \begin{rema}
%%%--- Note that a standard proof of Uryhson lemma may be represented as follows: iterate the lifting property to prove
%%%--- $$ \emptyset \longrightarrow X\rightthreetimes \{x \swarrow x_1 \searrow ... \swarrow x_n \searrow y \} \longrightarrow \{x \swarrow x_1 = ... = x_n \searrow y \}$$
%%%--- Then pass to the infinite limit to construct a map
%%%--- $ X \longrightarrow \mathbb{R}$.
%%%--- \end{rema}
%%%---
\subsection{\label{app:AEEA}Appendix. Compactness as being uniform: change of order of quantifiers}
We give several examples where an application of compactness
can be reformulated as changing the order of quantifiers in a formula.
\subsubsection{ %tex #######
Each real-valued function on a compact set is bounded
}
$$\frac{
\forall x \in K \exists M ( f(x) < M )
}{%------------------------------------
\exists M \forall x \in K ( f(x) < M )
}$$
%forall x in K exists M ( f(x) < M )
%------------------------------------
%exists M forall x in K ( f(x) < M )
%
Note this is a lifting property, for $K$ connected:
$$\{\}\longrightarrow K \rtt \sqcup_{ n\in\NN} (-n,n) \longrightarrow \RR$$
here $\sqcup_ n (-n,n) \longrightarrow \RR$ denotes the map to the real line
from the disjoint union of intervals $(-n,n)$ which cover it.
Note this is a standard example of an open covering of $\RR$ which
shows it is not compact.
\subsubsection{ %tex ###
The image of a closed set is closed
} %tex
%%%---
%%%--- We shall see that this can be understood as an instance of
%%%--- being uniform, i.e. a change of order of quantifiers $\AE\longrightarrow \EA$:
%%%---
$K$ is compact iff the following implication
holds for each set $X$ and each subset $Z \subset \XxK$:
%
$$\frac{
\forall y \in K \exists U \exists V ( U \subset X\text{ open and }V \subset K\text{ open and }a \in U\text{ and }y \in V\text{ and }\UxV \subset Z )
}{%---------------------------------------------------------------------------------------------------(K compact)
\exists U \exists V \forall y \in K ( U \subset X\text{ open and }V \subset K\text{ open and }a \in U\text{ and }y \in V\text{ and }\UxV \subset Z )
}$$
%
%
%
%forall y in K exists U exists V ( U \subset X open and V \subset K open and a in U and y in V and UxV \subset Z )
%---------------------------------------------------------------------------------------------------(K compact)
%exists U exists V forall y in K ( U \subset X open and V \subset K open and a in U and y in V and UxV \subset Z )
%
%
The hypothesis says $Z$ contains a rectangular open neighbourhood of each point of the line $\{a\}\times K$;
the conclusion says that $Z$ contains a rectangular open neighbourhood of the whole line $\{a\}\times K$.
\subsubsection{ %tex #######
A Hausdorff compact is necessarily normal.
} %tex
The application of compactness in the usual proof of this implication
amounts to the following change of order of quantifiers:
$$\frac{
\forall a \in A \forall b \in B \exists U \exists V ( a \in U\text{ and }b \in V\text{ and }U\cap V=\{\}\text{ and }U \subset K\text{ open and }V \subset K\text{ open} )
}{%----------------------------------------------------------------------------------------------------------------
\exists U \exists V \forall a \in A \forall b \in B ( a \in U\text{ and }b \in V\text{ and }U\cap V=\{\}\text{ and }U \subset K\text{ open and }V \subset K\text{ open} )}$$
%
%forall a in A forall b in B exists U exists V ( a in U and b in V and U/\V=\{\} and U \subset K open and V \subset K open )
%----------------------------------------------------------------------------------------------------------------
%exists U exists V forall a in A forall b in B ( a in U and b in V and U/\V=\{\} and U \subset K open and V \subset K open )
%
\subsubsection{ %tex #######
Lebesgue number Lemma
} %tex
Let $S$ be a family of (arbitrary) subsets of a metric space $X$.
$$\frac{
\forall x \in X \exists \delta>0 \exists U \in S \forall y \in X ( dist(x,y)<\delta \implies y \in U )
}{%------------------------------------------------------------------------------------------
\exists \delta>0 \forall x \in X \exists U \in S \forall y \in X ( dist(x,y)<\delta \implies y \in U )
}$$
%
%for each x in X exists delta>0 exists U in S forall y in X ( dist(x,y) y in U )
%------------------------------------------------------------------------------------------
%exists delta>0 for each x in X exists U in S forall y in X ( dist(x,y) y in U )
%
The hypothesis says that $\{ Inn\, U\ :\ U \in S \}$ is an open cover of $X$;
the conclusion is as usually stated, that each set of diameter $<\delta$ is covered by a single member of the cover.
%%fixme: shall i skip these (un)clarifications?
Note that this lemma may be expressed in terms of uniform structures.
\subsubsection{Paracompactness.}
\href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=6719&option_lang=eng}{[Alexandroff,\S2.3,p.38]}
writes {``as it seems to me, one of the deepest and most interesting
properties of paracompacts''} is the following theorem of A.Stone:
that \bqqq
A $T_1$-space is {\em paracompact} iff for each open covering $\alpha$ of $X$ there is an open covering $\beta$
such that for each $x$ in $X$ there is $U$ in $A$ such that
$\cup \{ V \in \beta : x \in V \} \subset U $
\eqqq
The family of subsets $\cup \{ V \in \beta : x \in V \}$ where $x\in X, \, V\in \beta$ forms a covering denoted by $\beta^*$
by [Alexandroff]. This is somewhat reminiscent of a simplicial construction.
As quantifier exchange, this is:
$$\frac{
\text{for each open covering }\alpha\text{ exists open covering }\beta.\ \forall x\in X\forall V \in \beta \exists U \in \alpha ( x \in V \implies V \subset U )
}{%--------------------------------------------------------------------------------------------------------------------
\text{for each open covering }\alpha\text{ exists open covering }\beta.\ \forall x\in X\exists U\in\alpha \forall V \in \beta ( x \in V \implies V \subset U )
}$$
%for each open covering A exists open covering B. for each x in X exists U in A for each V in A ( x in V ==> V \subset U )
%
%
%for each open covering A exists open covering $B$. for each x in X for each V in B exists U in A ( x in V ==> V \subset U )
%--------------------------------------------------------------------------------------------------------------------
%for each open covering A exists open covering B. for each x in X exists U in A for each V in A ( x in V ==> V \subset U )
%
The hypothesis holds trivially: take $\beta=\alpha , V=U$.
% Reformulate this property in simplicial terms.
%More generally, develop the theory of uniform structure in simplicial terms.
%%%alternatively:--
%%% Does Alexandroff imply that this is a simplicial property?
%%
%\end{enonce}
\begin{enonce}{Question} %tex #### Question/Reference Request.
Describe a logic and a class of formulae where such exchange of order
quantifiers is permissible. Is there a treatment of compactness in terms
of changing order of quantifiers ?
\end{enonce} %tex
\subsection*{Acknowledgements.} To be written.
This work is a continuation of [DMG]; early history is given there. I thank
M.Bays, G.Cherlin, D.Krachun, K.Pimenov, V.Sosnilo, S.Sinchuk and P.Zusmanovich for discussions and proofreading; I
thank L.Beklemishev, N.Durov, S.V.Ivanov, D.Grayson, 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.
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and the first (1914) German edition.
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\newblock Wilfrid Hodges.
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\newblock \url{http://wilfridhodges.co.uk/arabic05.pdf}
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\newblock David Holgate.
\newblock The pullback closure, perfect morphisms and completions.
\newblock PhD Thesis, University of Cape Town (1995).
\newblock \url{http://www.maths.uwc.ac.za/~dholgate/Papers/DBHPhd.zip}
\bibitem[Hocking, Young]{Hocking, Young}
\newblock John~G.~Hocking, Gail~S.~Young.
\newblock Topology.
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\newblock Yuri I. Manin.
\newblock Kolmogorov complexity as a hidden factor of scientific discourse: from Newton's law to data mining.
\newblock 19 pages.Talk at the Plenary Session of the Pontifical Academy of Sciences on "Complexity and Analogy in Science: Theoretical, Methodological and Epistemological Aspects", Casina Pio IV, Nov. 5-7, 2012. November 5--7, 2012
\newblock \url{https://arxiv.org/abs/1301.0081}
\bibitem[Stacks Project]{Stacts Project}
\newblock {The Stacks project}.
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\bibitem[Schupp]{Schupp}
\newblock P. Schupp.
\newblock A characterization of inner automorphisms.
\newblock Proceedings of the American Mathematical Society, vol. 101, n. 2,
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\bibitem[Inn]{Inn}
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\end{thebibliography}
\end{document}
\section{APPENDIX: old text, parts may be of use}
\subsubsection{Continuity--2nd WORSE? TRY.}
By \href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki, I\S2.1, Def.1, Def.2]},\newline\noindent
%%+ a mapping $f$ of a topological space $X$ into a topological
%%+ space $X'$ is said to be {\em continuous} at a point $x_0\in X$ if, given any neighbourhood $V'$
%%+ of $f (x_ 0)$ in $X'$, there is a neighbourhood $V$ of $x_ 0$ in $X$ such that the relation
%%+ $x \in V$ implies $f (x) \in V'$.
\includegraphics[width=\linewidth]{Bourbaki-Cont-func-IS2-1-Def1-p25-30.png}\newline\noindent
\includegraphics[width=\linewidth]{Bourbaki-Cont-func-IS2-1-Def1-Intuit-p26-31.png}\newline\noindent
\includegraphics[width=\linewidth]{Bourbaki-Cont-func-IS2-1-Def2-p26-31.png}\newline\noindent
\includegraphics[width=\linewidth]{Bourbaki-Cont-func-intro-Intuit-p13.png}\newline\noindent
%
{\sf %Definition I may be restated in the following more intuitive form: to say
%that f is continuous at the point Xo means that
\em $f(x)$ is as near as we please to $f (x_0)$}:
means pair $(f(x_0),f(x))$ lies in a given topoic subset of $X'\times X'$
{\sf\em whenever $x$ is sufficiently near $x_0$}:
whenever $(x_0,x)$ lies in a sufficiently small topoic subset of $X\times X$.
That is,
%%
%% This says that the preimage of a topoic subset $\{f(x_0)\}\times V'\cup (X'\setminus \{f(x_0)\}\times X$
%% contains a topoic subset $\{x_0\}\times V \cup (X\setminus\{x_0\})\times X$.
%% By \href{http://mishap.sdf.org/mints-lifting-property-as-negation/tmp/Bourbaki_General_Topology.djvu}{[Bourbaki,I\S2.1,Def.2]}, a mapping of a topological space X into a topological space X'
%% is said to be continuous on $X$ (or just continuous) if it is continuous at each point
%% of $X$, i.e. iff the preimage of a topoic subset of $X'\times X'$ is necessarily
%% a topoic subset of $X\times X$, i.e.
%%
iff the map $(f,f):X\times X \lra X'\times X'$ is continuous in the filter topology.
\section{Limits:OLD}
%% PROPOSITION I. A filter base
%% on a topological space X converges to x if
%% and onry if every set of a fundamental s.,vstem of neighbourhoods of x contain..r
%% a set of
%% .
%% In accordance with the terminology introduced in
%% 1, no. 2 we can state
%% Proposition 1 in the following way :
%% converges to x if and only if
%% there are sets of
%% as near as we please to x.
\includegraphics[width=\linewidth]{Bourbaki-limit-func-IS7-3-Def3-p70.png}\newline\noindent
\includegraphics[width=\linewidth]{Bourbaki-filter-converge-IS7-1-Prop1-p69.png}\newline\noindent
{\sf as near as we please to $x$} suggests we pick an arbitrary neighbourhood $U_x$ of $x$
%we pick an arbitrary topoic subset of $U\subset X\times X$;
{\sf there are sets of $\mathfrak B$ as near as we please to $x$} suggests $U_x\in \mathfrak B$.
Equivalently, for an arbitrary topoic subset of $\{x\}\times U_x\cup U' \subset X\times X$ where $U'\subset (X\setminus \{x\})\times X$,
it holds that $U_x\in \mathfrak B$, i.e. the map
$$
\mathfrak B \xra{x_*} X\times X, \ \ x'\longmapsto (x,x')$$
is continious in the filter topology. This extends to a map of simplicial objects:
$
\xymatrix{ {} & {\mathfrak B} \ar[r]|{\text{id}} \ar@{->}[d]|{(x,x')} & {\mathfrak B} \ar[r]|{\text{id}}\ar[l] \ar[d]|{(x,x',x')} & {\mathfrak B} \ar[r]|{\text{id}}\ar[l] \ar[d]|{(x,x',x',x')} & {} \ar[l] \\ {X} \ar[r] &{ X\times X } \ar[r]\ar[l] & { X\times X\times X } \ar[r]\ar[l] & { X\times X\times X\times X } \ar[r]\ar[l] & {} \ar[l] } $
Now, in self-explanotary notation, we see that {\em a filter $\mathfrak B$ on a topological space $X$ converges to $x$}
iff we are allowed to write
$$
x\times id: id(\mathfrak B) \xra{x_*} X[-1], \ \ x'\longmapsto (x,x',..,x')$$
% DEFINITION 3. Let f be a mapping of a set X into a topological space Y,
%% and let
%% be afilter on X. A point ye Y is said to be a limit point (or simply
%% a limit) (resp. cluster point) of f with respect to the filter
%% if y is a limit point
%% (resp. cluster point) of the filter base f (ij).
\newline\noindent \includegraphics[width=\linewidth]{Bourbaki-limit-func-IS7-3-Def3-p70.png}\newline\noindent
Thus, $y\in Y$ is a limit point of $f:X\lra Y$ with respect to the filter $\mathfrak F$
iff we can write $$
y\times id(f): id(\mathfrak F) \lra Y[-1], \ \ x'\longmapsto (y,x,..,x)$$
\section{1st Try based on Hatcher. Pathspaces and fibrations and shifts...}
\includegraphics[width=\linewidth]{hatcher-pathspace-fibr-full.png}
%%+
%%+ \begin{quote}
%%+ There is a simple but extremely useful way to turn arbitrary mappings into fi-
%%+ brations. Given a map /:A—<•£, let Ef be the space of pairs (a, y) where a e A
%%+ and y.I^-B is a path in B with y(0) = f(a). We topologize Ef as a subspace of
%%+ AxB1, where B1 is the space of mappings J—<•£ with the compact-open topology; see
%%+ the Appendix for the definition and basic properties of this topology, in particular
%%+ Proposition A.14 which we will be using shortly.
%%+ \end{quote}
%%+
An essential axiom of model categories is that each map decomposes as a composition
of a cofibration which is also a weak equivalence and a fibration.
In notation
$X \xra{(cw)} E \xra f Y$....
We want to understand this argument. .. Paths are complicated ..
To translate, we care both about intuition and algebraic manipulations present in the text.
We also stress the ``tautological'' character of the argument.
Intuitively, we only care about an infinitesimal neighbourhood of
$$\{(x,id_x):x\in X\}\subset E_f=\{(x,\gamma) \,:\, x\in X, \gamma:[0,1]\lra X, \gamma(0)=x\}.$$
So to say, we'd rather have our paths to be infinitesimally short.
The text mentions algebraic manipulations of considering pairs $(x, f(x))$
(we identify the constant path $id_x, id_x(t)=x$ with the point $x$;
well, in fact we ignore the distinction because we have no language to describe paths).
We also note that when $Y$ is a point, $E_f=X$, and when $X$ is a point,
$E_f$ is the space of paths starting at $x$.
Above suggests we consider the map $$X\lra X\times Y, \ x\mapsto (x,f(x)) $$
and we consider the simplicial complex associated to the space $X\times Y$
and the maps
\newline\noindent
$
\xymatrix{
{X} \ar[r]\ar[d]|{(x,f(x))} &{ X\times X } \ar[r]\ar[l]\ar[d]|{(x_1,f(x_1),x_2,f(x_2))} & { X\times X\times X } \ar[r]\ar[l]\ar[d]|{(x_1,f(x_1),x_2,f(x_2),x_3,f(x_3))} & {} \ar[l] \\
{X\times Y} \ar[r]\ar[d]|{y} &{ X\times Y\times X\times Y } \ar[r]\ar[l]\ar[d]|{(y_1,y_2)} & { X\times Y\times X\times Y\times X\times Y } \ar[r]\ar[l]\ar[d]|{(y_1,y_2,y_3)} & {} \ar[l] \\
{ Y} \ar[r] &{ Y\times Y } \ar[r]\ar[l] & { Y\times Y\times Y } \ar[r]\ar[l] & {} \ar[l] } \\
$
Now, to reflect the intuition that we only care about an infinitesimal neighbourhood of
$\{(x,id_x):x\in X\}$, we modify the filter on $X\times Y$ by making
``infinitesimal neighbourhoods of $X$''
$$\bigcup\limits_{x\in X,\, f(x)\in U_{f(x)}\text{ an open neighbourhood}}\,\{x\}\times U_{f(x)} \subset X\times Y$$ topoic, and similarly
for $(X\times Y)^n$.
This suggests a definition of Axiom M2 (cw)-(f) decomposition ; we may check that any (cw)-arrow so defined
does lift wrt any (f)-arrow so defined. SAY BETTER.
{\tt TODO: list of research problems, or rather directions}
\section{2nd try based on Novikov}
{\tt Kostya: maybe it's better to base our translation on the text of Novikov (below). It seems Novikov expresses the intuition more directly, and it's kinda more like what we do...}
\newline\noindent
\includegraphics[width=\linewidth]{Novikov-Fibration-Example.png}\newline\noindent
\includegraphics[width=\linewidth]{Novikov-Fibration-Example2.png}
....we consider the simplicial complex associated to the space $X\times Y$
and the maps
\newline\noindent
$
\xymatrix{
{B} \ar[r]\ar[d] &{ B\times B } \ar[r]\ar[l]\ar[d] & { B\times B\times B } \ar[r]\ar[l]\ar[d] & {} \ar[l] \\
{Y_B} \ar[r]\ar[d] &{ Y\times Y_B } \ar[r]\ar[l]\ar[d] & { Y\times Y\times Y_B } \ar[r]\ar[l]\ar[d] & {} \ar[l] \\
{ Y} \ar[r] &{ Y\times Y } \ar[r]\ar[l] & { Y\times Y\times Y } \ar[r]\ar[l] & {} \ar[l] } \\
$
Now, to reflect the intuition that we only care about an infinitesimal neighbourhood of $B$ in $X$ consiting of very short (better infinitesimally short) paths starting in $B$,
we modify the filter on $Y$ by making topoic all open subset of $Y$ containing $B$; for $Y^n$, we make topoic all subset of form $U\times U\times ..\times U$
where $B\subset U\subset Y$ is an open neighbourhood of $B$.
This suggests a definition of Axiom M2 (cw)-(f) decomposition ; we may check that any (cw)-arrow so defined
\newline\noindent\includegraphics[width=\linewidth]{Bourbaki-Axioms_V.png}
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
%References:
\end{document}
*