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

% «.defs»			(to "defs")
% «.title»			(to "title")
% «.abstract»			(to "abstract")
% «.inclusions»			(to "inclusions")
%
% «.yoneda»			(to "yoneda")
% «.and-and-imp»		(to "and-and-imp")
% «.canonical-subobjects»	(to "canonical-subobjects")
% «.subpoints»			(to "subpoints")

\documentclass[oneside,12pt,a4paper]{article}
%\documentclass[oneside,12pt,a5paper]{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")
%
\usepackage[backend=biber,
   style=alphabetic]{biblatex}            % (find-es "tex" "biber")
\addbibresource{catsem-slides.bib}        % (find-LATEX "catsem-slides.bib")
%
% (find-es "tex" "geometry")
\begin{document}

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

% %L dofile "edrxtikz.lua"  -- (find-LATEX "edrxtikz.lua")
% %L dofile "edrxpict.lua"  -- (find-LATEX "edrxpict.lua")
% \pu




\newpage






% «and-and-imp»  (to ".and-and-imp")
% (cljp 3 "and-and-imp")
% (clj     "and-and-imp")


\section{Conjunction and implication}


% «canonical-subobjects»  (to ".canonical-subobjects")
% (cljp 1 "canonical-subobjects")
% (clj    "canonical-subobjects")

\section{Canonical subobjects}

$\CanSub(E)$ is the set of canonical subobjects of $E$.

The notation $D⊆E$ means $D∈\CanSub(E)$.

If $D⊆E$ the canonical monic $D \monicto E$ is called an {\sl
  inclusion}.

Pullbacks of inclusions are inclusions.

All our arrows written as `$\monicto$' will be inclusions except where
explicitly indicated. Using inclusions (almost) everywhere will let us
use a set-theoretic notation for several operations -- for example, if
$C,D⊆E$ then $C∩D$ is their product in $\CanSub(E)$ (a pullback!) and
$C∪D$ is their coproduct. In diagrams:


%D diagram ??
%D 2Dx     100 +25 +25 +25 +25
%D 2D  100 A0  A1  B0  B1  B2
%D 2D
%D 2D  +25 A2  A3      B3
%D 2D
%D ren A0 A1 A2 A3 ==> C∩D D C E
%D ren B0 B1 B2 B3 ==> C C∪D D E
%D
%D (( A0 A1 >->
%D    A0 A2 >->
%D    A1 A3 >->
%D    A2 A3 >->
%D    A0 relplace 7 7 \pbsymbol{7}
%D
%D    B0 B1 >->
%D    B1 B2 <-<
%D    B0 B3 >->
%D    B1 B3 >->
%D    B2 B3 >->
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$



\msk

1 is the (given) terminal object.

A {\sl subterminal} is a canonical subobject of 1.

The truth-values of \cite{PH1} and \cite{PH2} are subterminals here.



% «subpoints»  (to ".subpoints")
\section{Subpoints}

A {\sl subpoint} of $E$ is a canonical subobject of $E$ that is
isomorphic to a subterminal. We write $\SubPoints(E)$ for the set of
subpoints of $E$. The notation $R::E$ is an abbreviation for
$R∈\SubPoints(E)$.



\msk



$E = \bigcup_{R::E} E_R$.

$D = \bigcup_{R::E} D_R$.

$D^E = \bigcup_{R::E} D_R {}^R$.

If $C⊆D⊆E$ and $R::E$ then $C_R ⊆ D_R$ and $C_R {}^R ⊆ D_R {}^R$.

If $C⊆D⊆E$ then $C^E = \bigcup_{R::E} C_R {}^R ⊆ \bigcup_{R::E} D_R {}^R = D^E$.

$$\begin{array}{rcl}
  D^{EE} &=& (D^E)^E \\
         &=& \bigcup_{R::E} (D^E)_R {}^R \\
         &=& \bigcup_{R::E} (\bigcup_{S::E} D_S {}^S)_R {}^R \\
         &=& \bigcup_{R::E} (\bigcup_{S::E} D_S {}^S {}_R)^R \\
         &=& \bigcup_{R::E} (\bigcup_{S::E} D_{R∩S} {}^{R∩S})^R \\
         &=& \bigcup_{R::E} (\bigcup_{Q::R} D_Q {}^Q)^R \\
         &=& \bigcup_{R::E} ((D_R)^R)^R \\
         &=& \bigcup_{R::E} D_R {}^R \\
         &=& D^E \\
  \end{array}
$$



%D diagram ??
%D 2Dx     100 +20 +20 +20 +20 +20 +20 +20
%D 2D  100 A0              B0
%D 2D  +20     A1              B1
%D 2D  +20 A2              B2
%D 2D
%D 2D  +20         C0              D0
%D 2D  +20             C1              D1
%D 2D  +20         C2              D2
%D 2D
%D ren A0 A1 A2 ==> D_{R∩S} D_{R∩S}{}^{R∩S} R∩S
%D ren B0 B1 B2 ==> D_S     D_S{}^S         S
%D ren C0 C1 C2 ==> D_R     D_R{}^R         R
%D ren D0 D1 D2 ==> D       D^E             E
%D
%D (( A0 A2 >-> A0 A1 >-> A1 A2 >->
%D    B0 B2 >-> B0 B1 >-> B1 B2 >->
%D    C0 C2 >-> C0 C1 >-> C1 C2 >->
%D    D0 D2 >-> D0 D1 >-> D1 D2 >->
%D    A2 B2 >->
%D    A2 C2 >->
%D    B2 D2 >->
%D    C2 D2 >->
%D
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$



%D diagram ??
%D 2Dx     100 +20 +20 +20 +40 +20 +20 +20 +20
%D 2D  100 A0      B0      C0      D0
%D 2D  +20     A1      B1      C1      D1
%D 2D  +20 A2      B2      C2      D2  D3
%D 2D
%D ren A0 A1 A2 ==> Q Q^* 1
%D ren B0 B1 B2 ==> P P^Q=P^*{∧}Q Q
%D ren C0 C1 C2 ==> R R^S S
%D ren D0 D1 D2 ==> {\Can}R {\Can}R^{{\Can}S} {\Can}S
%D
%D (( A0 A2 >-> .plabel= l q
%D    A0 A1 >->
%D    A1 A2 >-> .plabel= r \ovl{q}
%D
%D    B0 B2 >-> .plabel= l p
%D    B0 B1 >->
%D    B1 B2 >-> .plabel= r \ovl{p}
%D
%D    C0 C2 >-> .plabel= l r
%D    C0 C1 >->
%D    C1 C2 >-> .plabel= r \ovl{r}
%D
%D    D0 D2 >-> .plabel= l p
%D    D0 D1 >->
%D    D1 D2 >-> .plabel= r \ovl{p}
%D    
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$

$\Clo$


\newpage

% (larp 9 "21.2._lemma")
% (lar    "21.2._lemma")




\printbibliography

\end{document}

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%                        
% <make>

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

% Local Variables:
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