% (find-LATEX "2020mclarty.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2020mclarty.tex" :end))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2020mclarty.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2020mclarty.pdf"))
% (defun e () (interactive) (find-LATEX "2020mclarty.tex"))
% (defun u () (interactive) (find-latex-upload-links "2020mclarty"))
% (defun v () (interactive) (find-2a '(e) '(d)) (g))
% (find-pdf-page   "~/LATEX/2020mclarty.pdf")
% (find-sh0 "cp -v  ~/LATEX/2020mclarty.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2020mclarty.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2020mclarty.pdf
%               file:///tmp/2020mclarty.pdf
%           file:///tmp/pen/2020mclarty.pdf
% http://angg.twu.net/LATEX/2020mclarty.pdf
% (find-LATEX "2019.mk")

% «.13.3._conjunction»	(to "13.3._conjunction")
% «.21._topologies»	(to "21._topologies")
% «.21.1._theorem»	(to "21.1._theorem")
%   «.dense-and-closed»	(to "dense-and-closed")
% «.21.2._lemma»	(to "21.2._lemma")

\documentclass[oneside,12pt]{article}
\usepackage[colorlinks,citecolor=DarkRed,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref")
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
\usepackage[x11names,svgnames]{xcolor} % (find-es "tex" "xcolor")
%\usepackage{colorweb}                 % (find-es "tex" "colorweb")
%\usepackage{tikz}
%
% (find-dn6 "preamble6.lua" "preamble0")
\usepackage{proof}   % For derivation trees ("%:" lines)
\input diagxy        % For 2D diagrams ("%D" lines)
\xyoption{curve}     % For the ".curve=" feature in 2D diagrams
%
\usepackage{edrx15}               % (find-LATEX "edrx15.sty")
\input edrxaccents.tex            % (find-LATEX "edrxaccents.tex")
\input edrxchars.tex              % (find-LATEX "edrxchars.tex")
\input edrxheadfoot.tex           % (find-LATEX "edrxheadfoot.tex")
\input edrxgac2.tex               % (find-LATEX "edrxgac2.tex")
%
% (find-es "tex" "geometry")
\begin{document}

\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"}  % (find-LATEX "dednat6load.lua")

% (find-books "__cats/__cats.el" "mclarty")

{\setlength{\parindent}{0em}
\footnotesize

Notes on Colin McLarty's ``Elementary Categories, Elementary Toposes'' (1992).

\ssk

These notes are at:

\url{http://angg.twu.net/LATEX/2020mclarty.pdf}

See:

\url{http://angg.twu.net/LATEX/2020favorite-conventions.pdf}

\url{http://angg.twu.net/math-b.html\#favorite-conventions}


}

\bsk
\bsk


\newpage

%  _ _____  _____    ____             _                  _   _             
% / |___ / |___ /   / ___|___  _ __  (_)_   _ _ __   ___| |_(_) ___  _ __  
% | | |_ \   |_ \  | |   / _ \| '_ \ | | | | | '_ \ / __| __| |/ _ \| '_ \ 
% | |___) | ___) | | |__| (_) | | | || | |_| | | | | (__| |_| | (_) | | | |
% |_|____(_)____/   \____\___/|_| |_|/ |\__,_|_| |_|\___|\__|_|\___/|_| |_|
%                                  |__/                                    
%
% «13.3._conjunction»  (to ".13.3._conjunction")
% (larp 2 "13.3._conjunction")
% (lar    "13.3._conjunction")
% (find-books "__cats/__cats.el" "mclarty")
% (find-mclartypage (+ 4 118) "13.3. Conjunction and intersection")
\subsection*{13.3. Conjunction and intersection}

The arrow $t$ is sometimes called the {\sl generic sub-object} because
it is a sub-object of $Ω$ itself, and every sub-object is a pullback
of it along exactly one arrow. There is also a generic pair of
sub-objects, namely $t×Ω:1×Ω→Ω×Ω$ and $Ω×t:Ω×1→Ω×Ω$.

Theorem 13.2. Given any pair of sub-objects of an object $A$, $r:R
\monicto A$ and $s: S \monicto A$, there is a unique arrow $u:A→Ω×Ω$
that makes both the (lower) squares below pullbacks, and that arrow is
$u:=〈χ_r,χ_s〉$:
%
%D diagram ??
%D 2Dx     100  +25 +20  +25     +40  +30 +25  +30
%D 2D  100 A0 - A1  B0 - B1      a0 - a1  b0 - b1
%D 2D      |     |  |     |      |     |  |     |
%D 2D  +17 |     |  |     |      |    a3' |    b3'
%D 2D  +8  A2 - A3  B2 - B3      a2 - a3  b2 - b3
%D 2D
%D 2D  +20 C0 - C1  D0 - D1      c0 - c1  d0 - d1
%D 2D      |     |  |     |      |     |  |     |
%D 2D  +17 |     |  |     |      |    c3' |    d3'
%D 2D  +8  C2 - C3  D2 - D3      c2 - c3  d2 - d3
%D 2D
%D ren A0 A1 A2 A3 ==> R 1 A Ω
%D ren B0 B1 B2 B3 ==> S 1 A Ω
%D ren a0 a1 a3' a2 a3 ==> a|_R * ⊤ a R(a)
%D ren b0 b1 b3' b2 b3 ==> a|_S * ⊤ a S(a)
%D
%D (( A0 A1  -> .plabel= a !
%D    A0 A2 >-> .plabel= l r
%D    A1 A3 >-> .plabel= r t
%D    A2 A3  -> .plabel= a χ_r
%D    A0 relplace 7 7 \pbsymbol{7}
%D
%D    B0 B1  -> .plabel= a !  
%D    B0 B2 >-> .plabel= l s  
%D    B1 B3 >-> .plabel= r t  
%D    B2 B3  -> .plabel= a χ_s
%D    B0 relplace 7 7 \pbsymbol{7}
%D
%D    a0 a1  |->
%D    a0 a2  |->
%D    a1 a3' |->
%D    a2 a3  |->
%D
%D    b0 b1  |->
%D    b0 b2  |->
%D    b1 b3' |->
%D    b2 b3  |->
%D ))
%D
%D ren C0 C1 C2 C3 ==> R 1{×}Ω A Ω{×}Ω
%D ren D0 D1 D2 D3 ==> S Ω{×}1 A Ω{×}Ω
%D ren c0 c1 c3' c2 c3 ==> a|_R (*,S(a)) ⊤ a (R(a),S(a))
%D ren d0 d1 d3' d2 d3 ==> a|_S (R(a),*) ⊤ a (R(a),S(a))
%D
%D (( C0 C1  -> .plabel= a !
%D    C0 C2 >-> .plabel= l r
%D    C1 C3 >-> .plabel= r t{×}Ω
%D    C2 C3  -> .plabel= a u
%D    C0 relplace 7 7 \pbsymbol{7}
%D
%D    D0 D1  -> .plabel= a !  
%D    D0 D2 >-> .plabel= l s  
%D    D1 D3 >-> .plabel= r Ω{×}t  
%D    D2 D3  -> .plabel= a u
%D    D0 relplace 7 7 \pbsymbol{7}
%D
%D    c0 c1  |->
%D    c0 c2  |->
%D    c1 c3' |->
%D    c2 c3  |->
%D
%D    d0 d1  |->
%D    d0 d2  |->
%D    d1 d3' |->
%D    d2 d3  |->
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$

Proof. Consider the following diagram:
%
%D diagram ??
%D 2Dx     100  +30  +25    +40  +35  +40
%D 2D  100 C0 - C1 - C2     c0 - c1 - c2
%D 2D      |     |    |     |     |    |
%D 2D  +17 |     |    |     |    c4'  c5'
%D 2D  +8  C3 - C4 - C5     c3 - c4 - c5
%D 2D      |                |
%D 2D      |                |
%D 2D  +20 C6               c6
%D 2D                                         
%D 2D  +15 D0 - D1 - D2     d0 - d1 - d2
%D 2D      |     |    |     |     |    |
%D 2D  +17 |     |    |     |    d4'  d5'
%D 2D  +8  D3 - D4 - D5     d3 - d4 - d5
%D 2D      |                |
%D 2D      |                |
%D 2D  +20 D6               d6
%D 2D                                         
%D ren C0 C1 C2 C3 C4 C5 C6 ==> R 1{×}Ω 1   A Ω{×}Ω Ω   Ω
%D ren D0 D1 D2 D3 D4 D5 D6 ==> S Ω{×}1 1   A Ω{×}Ω Ω   Ω
%D ren c0 c1 c4' c2 c5'     ==> a|_R (*,S(a)) (⊤,S(a)) * ⊤
%D ren c3    c4     c5  c6  ==> a (R(a),S(a)) R(a)         S(a)
%D ren d0 d1 d4' d2 d5'     ==> a|_S (R(a),*) (R(a),⊤) * ⊤
%D ren d3    d4     d5  d6  ==> a (R(a),S(a)) S(a)         S(a)
%D
%D (( C0 C1  -> .plabel= a 〈!,χ_{s∘r}〉
%D    C1 C2  -> .plabel= a !
%D    C0 C3 >-> .plabel= l r
%D    C1 C4 >-> .plabel= m t{×}Ω
%D    C2 C5 >-> .plabel= r t
%D    C3 C4  -> .plabel= a 〈χ_r,χ_s〉
%D    C4 C5  -> .plabel= a p_1
%D    C3 C5  -> .plabel= b χ_r .slide= -10pt
%D    C3 C6  -> .plabel= l χ_s
%D    C0 relplace 7 7 \pbsymbol{7}
%D    C1 relplace 9 7 \pbsymbol{7}
%D 
%D    c0 c1 |->
%D    c1 c2 |->
%D    c0 c3 |->
%D    c1 c4' |->
%D    c2 c5' |->
%D    c3 c4 |->
%D    c4 c5 |->
%D    c3 c6 |->
%D ))
%D (( D0 D1  -> .plabel= a 〈χ_{r∘s},!〉
%D    D1 D2  -> .plabel= a !
%D    D0 D3 >-> .plabel= l s
%D    D1 D4 >-> .plabel= m Ω{×}t
%D    D2 D5 >-> .plabel= r t
%D    D3 D4  -> .plabel= a 〈χ_r,χ_s〉
%D    D4 D5  -> .plabel= a p_2
%D    D3 D5  -> .plabel= b χ_s .slide= -10pt
%D    D3 D6  -> .plabel= l χ_s
%D    D0 relplace 7 7 \pbsymbol{7}
%D    D1 relplace 9 7 \pbsymbol{7}
%D 
%D    d0 d1 |->
%D    d1 d2 |->
%D    d0 d3 |->
%D    d1 d4' |->
%D    d2 d5' |->
%D    d3 d4 |->
%D    d4 d5 |->
%D    d3 d6 |->
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$
%
The left-hand square is a pullback iff the outer rectangle is; that
is, iff $p_1∘u=χ_s$. Similarly, $p_2∘u=χ_r$.



\newpage



%  _  ___     _   _ _   _  ___      
% / |/ _ \   | \ | | \ | |/ _ \ ___ 
% | | (_) |  |  \| |  \| | | | / __|
% | |\__, |  | |\  | |\  | |_| \__ \
% |_|  /_(_) |_| \_|_| \_|\___/|___/
%                                   
% (find-mclartypage (+ 4 172) "19. Natural numbers objects")

% \section*{19. Natural numbers objects}





%  ____  _     _____                 _             _           
% |___ \/ |   |_   _|__  _ __   ___ | | ___   __ _(_) ___  ___ 
%   __) | |     | |/ _ \| '_ \ / _ \| |/ _ \ / _` | |/ _ \/ __|
%  / __/| |_    | | (_) | |_) | (_) | | (_) | (_| | |  __/\__ \
% |_____|_(_)   |_|\___/| .__/ \___/|_|\___/ \__, |_|\___||___/
%                       |_|                  |___/             
%
% «21._topologies»  (to ".21._topologies")
% (larp 3 "21._topologies")
% (lar    "21._topologies")
% (find-books "__cats/__cats.el" "mclarty")
% (find-mclartypage (+ 4 196) "21. Topologies")

\section*{21. Topologies}

A {\sl (Lawvere-Tierney) topology} on a topos is an arrow $j:Ω→Ω$ that
obeys:

(1) $j∘t=t$,

(2) $j∘j=j$,

(3) $j∘∧=∧∘(j×j)$.

In diagrams:
%
%D diagram ??
%D 2Dx     100 +20 +20 +20 +30 +25
%D 2D  100 A0  A1  B0  B1  C0  C1
%D 2D
%D 2D  +20     A2      B2  C2  C3
%D 2D
%D ren A0 A1 A2 ==> Ω Ω Ω
%D ren B0 B1 B2 ==> Ω Ω Ω
%D ren C0 C1 C2 C3 ==> Ω×Ω Ω Ω×Ω Ω
%D
%D (( A0 A1 -> .plabel= a t
%D    A0 A2 -> .plabel= l t
%D    A1 A2 -> .plabel= r j
%D
%D    B0 B1 -> .plabel= a j
%D    B0 B2 -> .plabel= l j
%D    B1 B2 -> .plabel= r j
%D
%D    C0 C1 -> .plabel= a ∧
%D    C0 C2 -> .plabel= l j×j
%D    C1 C3 -> .plabel= r j
%D    C2 C3 -> .plabel= a ∧
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$

% (tptp 2 "3.13._universal_closure")
% (tpt    "3.13._universal_closure")

A {\sl universal closure operation} on a topos is defined by
specifying, for each object $A$ of the topos, a closure operation
(i.e. an increasing, order-preserving, idempotent map) on the poset of
subobjects of $A$ in such a way that closure commutes with pullback
along morphisms. Let's denote the closure of $s:S \monicto X$ by
$\ovl{s}: \ovl{S} \monicto X$; more formally, the definition says that
for any subobjects $s$ and $w$ of $A$, and arrow $f:B→A$, we have:

(1) $s⊆\ovl{s}$,

(2) $\ovl{s}≡\ovl{\ovl{s}}$,

(3) $\ovl{(s∩w)} ≡ \ovl{s}∩\ovl{w}$,

(4) $s⊆w$ implies $\ovl{s}⊆\ovl{w}$, 

(5) $\ovl{f^{-1}(s)} = f^{-1}(\ovl{s})$.

\bsk


\newpage

% «21.1._theorem»  (to ".21.1._theorem")
% (larp 4 "21.1._theorem")
% (lar    "21.1._theorem")

Theorem 21.1. For any topology $j$, subobjects $s$ and $w$ of $A$, and
arrow $f:B→A$:

(1) $s⊆\ovl{s}$,

(2) $\ovl{s}≡\ovl{\ovl{s}}$,

(3) $\ovl{(s∩w)} ≡ \ovl{s}∩\ovl{w}$,

(4) if $s⊆w$ then $\ovl{s}⊆\ovl{w}$, 

(5) the $j$-closure of $f^{-1}(s)$ is $f^{-1}(\ovl{s})$.

We say the the $j$-closure operator is inflationary, idempotent, it
preserves intersections, it preserves order, and it is stable under
pullback.

\bsk

Here are the constructions.

(Thx to David Michael Roberts for helping me with item (1)!)

\msk

(1) The arrow $S \monicto \ovl{S}$ is a factorization through a
pullback:

%D diagram 21.1.(1)_inflationary
%D 2Dx     100  +20  100   +20 +20  +30  +20
%D 2D  100 A0 ------ A1 ------ A2 _____
%D 2D      |  \      |  \      |       \
%D 2D  +20 |    A3 --|--- A4 --|-------- A5
%D 2D      |  /      |  /      |       /
%D 2D  +20 A6 ------ A7 ------ A8 - A9
%D 2D
%D ren A0  A1  A2      ==> ? S 1
%D ren   A3  A4     A5 ==> ? \ovl{S} 1
%D ren A6  A7  A8 A9   ==> ? A Ω Ω
%D
%D (( # Horizontal arrows:
%D  # A0 A1 ->
%D    A1 A2 ->
%D  # A3 A4 ->
%D    A4 A5 ->
%D  # A6 A7 ->
%D    A7 A8 -> .plabel= b χ_s
%D    A8 A9 -> .plabel= b j
%D
%D    # Vertical arrows:
%D  # A0 A6 >->
%D    A1 A7 >-> .PLABEL= _(0.30) s
%D    A2 A8 >->
%D
%D    # Diagonal arrows:
%D  # A0 A3 >->
%D    A1 A4 >->
%D    A2 A5  ->
%D  # A3 A6 >->
%D    A4 A7 >-> .plabel= r \ovl{s}
%D    A5 A9 >->
%D
%D ))
%D enddiagram
%D
$$\pu
  \diag{21.1.(1)_inflationary}
$$

(2) Idempotent: we have $\ovl{\ovl{s}} = (j∘j∘χ_s)^{-1}(t) =
  (j∘χ_s)^{-1}(t) = \ovl{s}$, so $\ovl{\ovl{s}}$ and $\ovl{s}$ are the
  same subobject and $\ovl{S} \monicto \ovl{\ovl{S}}$ is the identity
  map.

%D diagram ??
%D 2Dx     100 +20 +15 +10 +20 +20 +35
%D 2D  100 A0 -------- A1
%D 2D  +15     A2 ------------ A3
%D 2D  +20         A4 ------------ A5
%D 2D
%D 2D  +15 A6 -------- A7  A8  A9
%D 2D
%D ren A0 A1 ==> S 1
%D ren A2 A3 ==> \ovl{S} 1
%D ren A4 A5 ==> \ovl{\ovl{S}} 1
%D ren A6 A7 A8 A9 ==> A Ω Ω Ω
%D
%D (( A0 A1 -> A0 A6 >-> .plabel= l s             A1 A7 >->
%D    A2 A3 -> A2 A6 >-> .plabel= l \ovl{s}       A3 A8 >->
%D    A4 A5 -> A4 A6 >-> .plabel= l \ovl{\ovl{s}} A5 A9 >->
%D    A0 A2 >-> A1 A3 >->
%D    A2 A4 >-> A3 A5 >->
%D    A6 A7 -> .plabel= b χ_s
%D    A7 A8 -> .plabel= b j
%D    A8 A9 -> .plabel= b j
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$

\newpage

(3) Preserves intersections:
%
%D diagram ??
%D 2Dx     100 +30 +30   +25 +40 +50
%D 2D  100 A0  A1  A2    B0  B1  B2
%D 2D  +22                       B4'
%D 2D   +8     A3  A4        B3  B4
%D 2D
%D ren A0 A1 A2 A3 A4 ==> A Ω×Ω Ω Ω×Ω Ω
%D ren B0 B1 B2 B4'   ==> a (P(a),Q(a)) P(a)∧Q(a) \ovl{P(a)∧Q(a)}
%D ren       B3 B4    ==>   (\ovl{P(a)},\ovl{Q(a)}) \ovl{P(a)}∧\ovl{Q(a)}
%D
%D (( A0 A1 -> .plabel= a 〈χ_s,χ_w〉
%D    A1 A2 -> .plabel= a ∧
%D    A1 A3 -> .plabel= l j×j
%D    A2 A4 -> .plabel= r j
%D    A3 A4 -> .plabel= a ∧
%D
%D    B0 B1  |->
%D    B1 B2  |->
%D    B1 B3  |->
%D    B2 B4' |->
%D    B3 B4  |->
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$

$$\begin{array}{rcl}
  j∘∧∘〈χ_s,χ_w〉 &=& j∘χ_{s∩w} \\
                &=& χ_{\ovl{s∩w}} \\
  j∘∧∘〈χ_s,χ_w〉 &=& ∧∘(j×j)∘〈χ_s,χ_w〉 \\
                &=& ∧∘〈j∘χ_s,j∘χ_w〉 \\
                &=& ∧∘〈χ_{\ovl{s}},χ_{\ovl{w}}〉 \\
                &=& χ_{\ovl{s}∩\ovl{w}} \\
  \end{array}
$$


(4) Preserves order:

%:
%:  s⊆w                          P≤Q
%:  -----                        -----
%:  s=s∩w                        P=P∧Q
%:  -----------------            -----------------
%:  \ovl{s}=\ovl{s∩w}            \ovl{P}=\ovl{P∧Q}
%:  -----------------------      -----------------------
%:  \ovl{s}=\ovl{s}∩\ovl{w}      \ovl{P}=\ovl{P}∧\ovl{Q}
%:  -----------------------      -----------------------
%:  \ovl{s}⊆\ovl{w}              \ovl{P}≤\ovl{Q}
%:
%:  ^pres-order-1                ^pres-order-2
%:
\pu
$$\ded{pres-order-1} \qquad \qquad \ded{pres-order-2}$$

\bsk
\bsk

(5) Stable under pullback. Here $W$ is a subobject of $B$, not of $A$:

%D diagram 21.1.(5)_stable_under_pullback
%D 2Dx     100  +25  +25   +25 +25  +30  +25
%D 2D  100 A0 ------ A1 ------ A2 _____
%D 2D      |  \      |  \      |       \
%D 2D  +25 |    A3 --|--- A4 --|-------- A5
%D 2D      |  /      |  /      |       /
%D 2D  +25 A6 ------ A7 ------ A8 - A9
%D 2D
%D ren A0  A1  A2      ==> W S 1
%D ren   A3  A4     A5 ==> \ovl{W} \ovl{S} 1
%D ren A6  A7  A8 A9   ==> B A Ω Ω
%D
%D (( # Horizontal arrows:
%D    A0 A1 ->
%D    A1 A2 ->
%D    A3 A4 ->
%D    A4 A5 ->
%D    A6 A7 -> .plabel= b f
%D    A7 A8 -> .plabel= b χ_s
%D    A8 A9 -> .plabel= b j
%D
%D    # Vertical arrows:
%D    A0 A6 >-> .PLABEL= _(0.30) \sm{w:=\\f^{-1}(s)}
%D    A1 A7 >-> .PLABEL= _(0.30) s
%D    A2 A8 >->
%D
%D    # Diagonal arrows:
%D    A0 A3 >->
%D    A1 A4 >->
%D    A2 A5  ->
%D    A3 A6 >-> .PLABEL= ^(0.20) \sm{\ovl{f^{-1}(s)}=\\f^{-1}(\ovl{s})}
%D    A4 A7 >-> .plabel= r \ovl{s}
%D    A5 A9 >->
%D
%D ))
%D enddiagram
%D
$$\pu
  \diag{21.1.(5)_stable_under_pullback}
$$


\newpage

\section*{Some consequences of stability by pullbacks}

{\bf Theorem.} If $α⊆β$ are subobjects of $C$ with mediating map $ι:A
\monicto B$, as in the first triangle below, then we have $\ovl{ι} =
\ovl{β^{-1}(α)} = β^{-1}(\ovl{α})$. If $α:A \monicto C$ and $β:B
\monicto C$ then the domain of $\ovl{ι}=β^{-1}(\ovl{α})$ is
$\ovl{A}∩B$. Proof:
%
%D diagram cor-tri-1
%D 2Dx     100 +15 +15
%D 2D  100 A       B
%D 2D
%D 2D  +25     C
%D 2D
%D # ren ==>
%D
%D (( A B >-> .plabel= a ι
%D    A C >-> .plabel= l α
%D    B C >-> .plabel= r β
%D ))
%D enddiagram
%D
%D diagram cor-tris-1
%D 2Dx     100 +20 +20 +20 +20 +20 +20
%D 2D  100 A0      B0      C0 (C0) 
%D 2D      |  \    |  \        |  \
%D 2D  +20 |   A1  |   B1      |   C1
%D 2D      |  /    |  /        |  /
%D 2D  +20 A2      B2          C2 
%D 2D
%D ren A0 A1 A2 ==> A \ovl{A} C
%D ren B0 B1 B2 ==> B \ovl{B} C
%D ren C0 C1 C2 ==> A \ovl{A}∩B B
%D
%D (( A0 A2 >-> .plabel= l α
%D    A0 A1 >->
%D    A1 A2 >-> .plabel= r \ovl{α} 
%D
%D    B0 B2 >-> .plabel= l β
%D    B0 B1 >->
%D    B1 B2 >-> .plabel= r \ovl{β} 
%D
%D    C0 C2 >-> .plabel= l ι
%D    C0 C1 >->
%D    C1 C2 >-> .plabel= r \ovl{ι}=β^{-1}(\ovl{α})  # \sm{\ovl{ι}=β^{-1}(α)\\\text{(Why???)}}
%D ))
%D enddiagram
%D
$$\pu
  \diag{cor-tri-1}
  \qquad
  \quad
  \diag{cor-tris-1}
$$
%
%D diagram 21.1._corollary_1
%D 2Dx     100 +40  +25  +25   +25 +25  +30  +25
%D 2D  100 L0  A0 ------ A1 ------ A2 _____
%D 2D          |  \      |  \      |       \
%D 2D  +25     |    A3 --|--- A4 --|-------- A5
%D 2D          |  /      |  /      |       /
%D 2D  +25     A6 ------ A7 ------ A8 - A9
%D 2D
%D ren L0 A0  A1  A2      ==> A A∩B A 1
%D ren      A3  A4     A5 ==>   \ovl{A}∩B \ovl{A} 1
%D ren    A6  A7  A8 A9   ==>   B C Ω Ω
%D
%D (( L0 A0 =
%D    
%D    # Horizontal arrows:
%D    A0 A1 >->
%D    A1 A2  ->
%D    A3 A4  ->
%D    A4 A5  ->
%D    A6 A7 >-> .plabel= b β
%D    A7 A8  -> .plabel= b χ_α
%D    A8 A9  -> .plabel= b j
%D
%D    # Vertical arrows:
%D    L0 A6 >-> .PLABEL= _(0.25) ι
%D    A0 A6 >-> .PLABEL= _(0.30) \sm{ι=\\β^{-1}(α)}
%D    A1 A7 >-> .PLABEL= _(0.30) α
%D    A2 A8 >->
%D
%D    # Diagonal arrows:
%D    A0 A3 >->
%D    A1 A4 >->
%D    A2 A5  ->
%D    A3 A6 >-> .PLABEL= ^(0.20) \sm{\ovl{ι}=\ovl{β^{-1}(α)}\\=β^{-1}(\ovl{α})}
%D    A4 A7 >-> .plabel= r \ovl{α}
%D    A5 A9 >->
%D
%D ))
%D enddiagram
%D
$$\pu
  \diag{21.1._corollary_1}
$$


{\bf First corollary.} Take any monic $s:S \monicto A$; its closure is
$\ovl{s}:\ovl{S} \monicto A$. Call its mediating map $ι:S \monicto
\ovl{S}$. Then $\ovl{ι} = \id_{\ovl{S}}: \ovl{S} \monicto \ovl{S}$.
Proof:
%
%D diagram stability_corollary_dense
%D 2Dx     100 +40  +25  +25   +25 +25  +30  +25
%D 2D  100 L0  A0 ------ A1 ------ A2 _____
%D 2D          |  \      |  \      |       \
%D 2D  +25     |    A3 --|--- A4 --|-------- A5
%D 2D          |  /      |  /      |       /
%D 2D  +25     A6 ------ A7 ------ A8 - A9
%D 2D
%D ren L0 A0  A1  A2      ==> S S∩\ovl{S} S 1
%D ren      A3  A4     A5 ==>   \ovl{S}∩\ovl{S}=\ovl{S} \ovl{S} 1
%D ren    A6  A7  A8 A9   ==>   \ovl{S} A Ω Ω
%D
%D (( L0 A0 =
%D    
%D    # Horizontal arrows:
%D    A0 A1 >->
%D    A1 A2  ->
%D    A3 A4  ->
%D    A4 A5  ->
%D    A6 A7 >-> .plabel= b \ovl{s}
%D    A7 A8  -> .plabel= b χ_s
%D    A8 A9  -> .plabel= b j
%D
%D    # Vertical arrows:
%D    L0 A6 >-> .PLABEL= _(0.25) ι
%D    A0 A6 >-> .PLABEL= _(0.30) \sm{ι=\\\ovl{s}^{-1}(s)}
%D    A1 A7 >-> .PLABEL= _(0.30) s
%D    A2 A8 >->
%D
%D    # Diagonal arrows:
%D    A0 A3 >->
%D    A1 A4 >->
%D    A2 A5  ->
%D    A3 A6 >-> .PLABEL= ^(0.20) \sm{\ovl{ι}=\ovl{\ovl{s}^{-1}(s)}=\\\ovl{s}^{-1}(\ovl{s})=\id}
%D    A4 A7 >-> .plabel= r \ovl{s}
%D    A5 A9 >->
%D
%D ))
%D enddiagram
%D
$$\pu
  \diag{stability_corollary_dense}
$$

\newpage

% «dense-and-closed»  (to ".dense-and-closed")
% (larp 7 "dense-and-closed")
% (lar    "dense-and-closed")

{\bf Two definitions: dense and closed.} Take a monic $s:S \monicto
A$. We say that $s$ is {\sl dense} when $\ovl{s}=\id_A$, i.e., when
$\ovl{S}=A$. And we say that $s$ is {\sl closed} when $\ovl{s}=s$,
i.e., when $\ovl{S}=S$.

For any monic $s:S \monicto A$ with mediating map $ι:S
\monicto \ovl{S}$ this mediating map is dense (by the First Corollary)
and $\ovl{s}$ is closed (because $\ovl{\ovl{s}} = \ovl{s}$). In a diagram:
%
%D diagram dense-closed
%D 2Dx     100 +20
%D 2D  100 A0
%D 2D
%D 2D  +20     A1
%D 2D
%D 2D  +20 A2
%D 2D
%D ren A0 A1 A2 ==> S \ovl{S} A
%D
%D (( A0 A1 >-> .plabel= r \isdense
%D    A0 A2 >-> .plabel= l s
%D    A1 A2 >-> .plabel= r \isclosed
%D ))
%D enddiagram
%D
$$\pu
  \def\isdense {\sm{ι\text{ is dense:}\\\ovl{ι}=\id_{\ovl{S}}}}
  \def\isclosed{\sm{\ovl{s}\text{ is closed:}\\\ovl{\ovl{s}}=\ovl{S}}}
  \diag{dense-closed}
$$

Also:

\ssk

If $s$ is closed the any $f^{-1}(s)$ is closed.

Proof: $s=\ovl{s}$, then $\ovl{f^{-1}(s)} = f^{-1}(\ovl{s}) =
f^{-1}(s)$.

\ssk

If $s$ is dense then any $f^{-1}(s)$ is dense.

Proof: $\ovl{s}=\id$, then $f^{-1}(\ovl{s}) = f^{-1}(\id) = \id$.

\ssk

If $s$ is dense and $s⊆w$ then $w$ is dense.

Proof: $\ovl{s}=\id$, so $\ovl{s}⊆\ovl{w}$, $\id⊆\ovl{w}$,
$\ovl{w}=\id$.

\ssk

If $s$ is dense and closed then $s=\id$.

Proof: $\ovl{s}=\id$ and $\ovl{s}=s$, so $s=\id$.



\newpage

{\bf Second corollary.} Let $p: P \monicto 1$, $q: Q \monicto 1$, and
$p⊆q$, with mediating map $ι:P \monicto Q$. Then $\ovl{i} =
q^{-1}(\ovl{p}): \ovl{P}∧Q \monicto Q$. Proof:


%D diagram stability_second_corollary
%D 2Dx     100 +40  +25  +25   +25 +25  +30  +25
%D 2D  100 L0  A0 ------ A1 ------ A2 _____
%D 2D          |  \      |  \      |       \
%D 2D  +25     |    A3 --|--- A4 --|-------- A5
%D 2D          |  /      |  /      |       /
%D 2D  +25     A6 ------ A7 ------ A8 - A9
%D 2D
%D ren L0 A0  A1  A2      ==> P P∧Q P 1
%D ren      A3  A4     A5 ==>   \ovl{P}∧Q \ovl{P} 1
%D ren    A6  A7  A8 A9   ==>   Q 1 Ω Ω
%D
%D (( L0 A0 =
%D    
%D    # Horizontal arrows:
%D    A0 A1 >->
%D    A1 A2  ->
%D    A3 A4  ->
%D    A4 A5  ->
%D    A6 A7 >-> .plabel= b q
%D    A7 A8  -> .plabel= b χ_p
%D    A8 A9  -> .plabel= b j
%D
%D    # Vertical arrows:
%D    L0 A6 >-> .PLABEL= _(0.25) ι
%D    A0 A6 >-> .PLABEL= _(0.30) \sm{ι=\\q^{-1}(p)}
%D    A1 A7 >-> .PLABEL= _(0.30) p
%D    A2 A8 >->
%D
%D    # Diagonal arrows:
%D    A0 A3 >->
%D    A1 A4 >->
%D    A2 A5  ->
%D    A3 A6 >-> .PLABEL= ^(0.20) \sm{\ovl{ι}=\ovl{q^{-1}(p)}\\=q^{-1}(\ovl{p})}
%D    A4 A7 >-> .plabel= r \ovl{p}
%D    A5 A9 >->
%D
%D ))
%D enddiagram
%D
$$\pu
  \diag{stability_second_corollary}
$$





% Template:
%
% %D diagram ??
% %D 2Dx     100  +20  +20   +20 +20  +30  +20
% %D 2D  100 A0 ------ A1 ------ A2 _____
% %D 2D      |  \      |  \      |       \
% %D 2D  +20 |    A3 --|--- A4 --|-------- A5
% %D 2D      |  /      |  /      |       /
% %D 2D  +20 A6 ------ A7 ------ A8 - A9
% %D 2D
% %D ren A0  A1  A2      ==> ? S 1
% %D ren   A3  A4     A5 ==> ? \ovl{S} 1
% %D ren A6  A7  A8 A9   ==> ? A Ω Ω
% %D
% %D (( # Horizontal arrows:
% %D    A0 A1 ->
% %D    A1 A2 ->
% %D    A3 A4 ->
% %D    A4 A5 ->
% %D    A6 A7 ->
% %D    A7 A8 -> .plabel= b χ_s
% %D    A8 A9 -> .plabel= b j
% %D
% %D    # Vertical arrows:
% %D    A0 A6 >->
% %D    A1 A7 >-> .PLABEL= _(0.30) s
% %D    A2 A8 >->
% %D
% %D    # Diagonal arrows:
% %D    A0 A3 >->
% %D    A1 A4 >->
% %D    A2 A5  ->
% %D    A3 A6 >->
% %D    A4 A7 >-> .plabel= r \ovl{s}
% %D    A5 A9 >->
% %D
% %D ))
% %D enddiagram
% %D
% $$\pu
%   \diag{??}
% $$


% (find-fline "~/IMAGES/mclarty_theorem_21.1_p1.png")
% (find-fline "~/IMAGES/mclarty_theorem_21.1_p2.png")



\newpage

% «21.2._lemma»  (to ".21.2._lemma")
% (larp 9 "21.2._lemma")
% (lar    "21.2._lemma")


% (find-books "__cats/__cats.el" "mclarty")
% (find-mclartypage (+ 4 196) "21. Topologies")


{\bf Lemma 21.2.} Suppose that $w$ is dense and $s$ is closed, and
that the square at the left below commutes. Then there is an arrow
$u:B→S$ making the two triangles commute, and this $u$ is unique.
%
%D diagram ??
%D 2Dx     100 +20 +25 +30 +25 +35 +30 +20 +25
%D 2D  100 A0          B0          C0
%D 2D
%D 2D  +20     A1  A2      B1  B2      C1  C2
%D 2D
%D 2D  +35     A3  A4      B3  B4      C3  C4
%D 2D
%D ren A0    A2 A3 A4 ==> W           S B A
%D ren B0 B1 B2 B3 B4 ==> W f^{-1}(S) S B A
%D ren C0 C1 C2 C3 C4 ==> W f^{-1}(S) S B A
%D
%D (( A0 A2  -> .plabel= a g
%D    A0 A3 >-> .plabel= l \sm{w\\\text{dense}}
%D    A2 A4 >-> .PLABEL= ^(0.30) \sm{s\\\text{closed}}
%D    A3 A4  -> .plabel= b f
%D
%D    B0 B1 >-> .plabel= m v=w
%D    B0 B2  -> .plabel= a g
%D    B0 B3 >-> .plabel= l \sm{w\\\text{dense}}
%D    B1 B2  -> .plabel= b u
%D    B1 B3 >-> .plabel= r \sm{f^{-1}(s)\\\text{closed}\\\text{dense}\\=\id}
%D    B2 B4 >-> .PLABEL= ^(0.30) \sm{s\\\text{closed}}
%D    B3 B4  -> .plabel= b f
%D
%D  # C0 C1  -> .plabel= m v=w
%D    C0 C2  -> .plabel= a g
%D    C0 C3 >-> .plabel= l \sm{w\\\text{dense}}
%D  # C1 C2  -> .plabel= b u
%D  # C1 C3  -> .plabel= r \sm{f^{-1}(s)\\\text{closed}\\\text{dense}\\=\id}
%D    C2 C4 >-> .PLABEL= ^(0.30) \sm{s\\\text{closed}}
%D    C3 C4  -> .plabel= b f
%D    C3 C2  -> .plabel= a u
%D    
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$

Proof. Build the pullback of $s$ along $f$ and call its upper arrow
$u$. As $s$ is closed, we have that $f^{-1}(s)$ is closed. Build
$v:W→f^{-1}(S)$ by factorization through this pullback; this $v$ is a
monic. The arrow $w$ is dense and $w⊆f^{-1}(s)$, so $w⊆f^{-1}(s)$ is
dense; as $f^{-1}(s)$ is dense and closed it is the identity, so
$v=w$, and the composite $B \to f^{-1}(S) \to S$ is $u$.

I don't know how to prove that this $u$ is unique.

% (find-books "__cats/__cats.el" "borceux")
% (find-borceux1page (+ 17 209) "5.5 Factorization systems")





\end{document}

%  __  __       _        
% |  \/  | __ _| | _____ 
% | |\/| |/ _` | |/ / _ \
% | |  | | (_| |   <  __/
% |_|  |_|\__,_|_|\_\___|
%                        
% <make>

 (eepitch-shell)
 (eepitch-kill)
 (eepitch-shell)
# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2020mclarty veryclean
make -f 2019.mk STEM=2020mclarty pdf

% Local Variables:
% coding: utf-8-unix
% ee-tla: "lar"
% End:
