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% (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:
% coding: utf-8-unix
% ee-tla: "clj"
% End: