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% (find-LATEX "2020lindenhovius.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2020lindenhovius.tex" :end))
% (defun C () (interactive) (find-LATEXSH "lualatex 2020lindenhovius.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2020lindenhovius.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2020lindenhovius.pdf"))
% (defun e () (interactive) (find-LATEX "2020lindenhovius.tex"))
% (defun u () (interactive) (find-latex-upload-links "2020lindenhovius"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (defun d0 () (interactive) (find-ebuffer "2020lindenhovius.pdf"))
% (code-eec-LATEX "2020lindenhovius")
% (find-pdf-page "~/LATEX/2020lindenhovius.pdf")
% (find-sh0 "cp -v ~/LATEX/2020lindenhovius.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2020lindenhovius.pdf /tmp/pen/")
% file:///home/edrx/LATEX/2020lindenhovius.pdf
% file:///tmp/2020lindenhovius.pdf
% file:///tmp/pen/2020lindenhovius.pdf
% http://angg.twu.net/LATEX/2020lindenhovius.pdf
% (find-LATEX "2019.mk")
%
% «.defs» (to "defs")
\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")
\input 2017planar-has-defs.tex % (find-LATEX "2017planar-has-defs.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 forths["<.>"] = function () pusharrow("<.>") end
%L forths["<-->"] = function () pusharrow("<-->") end
%L forths["|-->"] = function () pusharrow("|-->") end
%L forths["<--|"] = function () pusharrow("<--|") end
% %L dofile "edrxtikz.lua" -- (find-LATEX "edrxtikz.lua")
% %L dofile "edrxpict.lua" -- (find-LATEX "edrxpict.lua")
% \pu
% «defs» (to ".defs")
\def\bfP{\mathbf{P}}
\def\Ups{\mathsf{U}}
\def\Downs{\mathsf{D}}
\def\Filts{\mathsf{F}}
\def\Jcan{{J_\mathrm{can}}}
\def\hasmax{\mathsf{hasmax}}
\def\trans {\mathsf{trans}}
\def\stab {\mathsf{stab}}
\def\plarray#1{\left(\begin{array}{l}#1\end{array}\right)}
\def\setofsc#1#2{\{\,#1\;:\;#2\,\}}
\def\Sieveson{\mathsf{Sieves\_on}}
\def\Coveringsieveson{\mathsf{Covering\_sieves\_on}}
\def\Coveringsieveson{\mathsf{Covsieves\_on}}
\def\Int{\mathsf{Int}}
\def\GrTops{\mathsf{GrTops}}
\def\Nucs{\mathsf{Nucs}}
\def\nuc{(·)^*}
\def\OX{\Opens(X)}
\def\OH{\Opens(H)}
\def\OB{\Opens(B)}
\def\OU{\Opens(U)}
\def\OV{\Opens(V)}
\def\catD{{\mathbf{D}}}
\def\catN{{\mathbf{N}}}
\def\calM{{\mathcal{M}}}
\def\calY{{\mathcal{Y}}}
\def\calT{{\mathcal{T}}}
\def\calH{{\mathcal{H}}}
\def\SetD{{\Set^\catD}}
\def\DP {\calD(\bfP)}
\def\Ddp{\calD({↓}p)}
\def\GP{\calG(\bfP)}
\def\Ddp{\calD({↓}p)}
\def\Nuc{\mathrm{Nuc}}
\def\Con{\mathrm{Con}}
\def\NucDP{\Nuc(\DP)}
\def\ConDP{\Con(\DP)}
\def\Onep {1${}'$}
\def\Onepp{1${}''$}
% «House» (to ".House")
%
%R local house, ohouse = 2/ #1 \, 7/ !h11111 \
%R |#2 #3| | !h01111 |
%R \#4 #5/ | !h01011 !h00111 |
%R |!h01010 !h00011 !h00101|
%R | !h00010 !h00001 |
%R \ !h00000 /
%R local houser = 1/ 1 \
%R |2 3|
%R \4 5/
%R
%R house:tomp({def="zfHouse#1#2#3#4#5", scale="6pt", meta="s"}):addcells():output()
%R house:tomp({zdef="House" , scale="20pt", meta=nil}):addbullets():addarrows():output()
%R houser:tomp({zdef="House" , scale="25pt", meta=nil}):addcells():addarrows():output()
%R ohouse:tomp({zdef="OHouse", scale="32pt", meta=nil}):addcells():addarrows("w"):output()
\pu
% «Bottle» (to ".Bottle")
% (find-LATEX "2021groth-tops-defs.tex" "Bottle")
% (find-angg "LUA/defwithmanyargs.lua" "SetManyArgs-tests")
% (find-angg "LUA/defwithmanyargs.lua" "SetManyArgs-tests" "Bottle")
%
\def\sa#1#2{\expandafter\def\csname myarg#1\endcsname{#2}}
\def\ga#1{\csname myarg#1\endcsname}
%
\makeatletter
\def\BottleSetArgs#1{\BottleSetArgs@#1}
\def\BottleSetArgs@#1#2#3#4#5{%
\sa{32}{#1}\sa{20}{#2}\sa{21}{#3}\sa{22}{#4}\sa{10}{#5}%
\BottleSetArgs@@}
\def\BottleSetArgs@@#1#2#3#4#5{%
\sa{11}{#1}\sa{12}{#2}\sa{00}{#3}\sa{01}{#4}\sa{02}{#5}%
}
\makeatother
%
%R local Bottle = 7/ !ga{32} \
%R | !ga{22} |
%R | !ga{21} !ga{12} |
%R |!ga{20} !ga{11} !ga{02}|
%R | !ga{10} !ga{01} |
%R \ !ga{00} /
%R Bottle:tomp({zdef="Bottle-5pt", scale="5pt", meta="s"}):addcells():output()
%R Bottle:tomp({zdef="Bottle-6pt", scale="6pt", meta="s"}):addcells():output()
%R Bottle:tomp({zdef="Bottle-8pt", scale="8pt", meta="s"}):addcells():output()
%R Bottle:tomp({zdef="Bottle^2", scale="52pt", meta=nil}):addcells():addarrows():output()
\pu
\def\bo #1{{ \BottleSetArgs{#1}\zha{Bottle-5pt} }}
\def\bbo #1{{\left[ \BottleSetArgs{#1}\zha{Bottle-5pt} \right]}}
\def\pwbo#1{{\left( \BottleSetArgs{#1}\zha{Bottle-8pt} \right)}}
% % Tests:
% $\bo{0 123 456 789} \bbo{0 123 456 789} \pwbo{· {20}{21}· {10}{11}· {00}{01}·}$
%
% $$Ω =
% \left(
% \BottleSetArgs{
% {\bbo{? ??? ??? ???}}
% {\bbo{· ?·· ?·· ?··}} {\bbo{· ??· ??· ??·}} {\bbo{· ??? ??? ???}}
% {\bbo{· ··· ?·· ?··}} {\bbo{· ··· ??· ??·}} {\bbo{· ··· ??? ???}}
% {\bbo{· ··· ··· ?··}} {\bbo{· ··· ··· ??·}} {\bbo{· ··· ··· ???}}}
% \zha{Bottle^2}
% \right)
% $$
% «WideBottle» (to ".WideBottle")
% (find-LATEX "2021groth-tops-defs.tex" "WideBottle")
% (find-angg "LUA/defwithmanyargs.lua" "SetManyArgs-tests")
% (find-angg "LUA/defwithmanyargs.lua" "SetManyArgs-tests" "WideBottle")
\makeatletter
\def\WideBottleSetArgs#1{\WideBottleSetArgs@#1}
\def\WideBottleSetArgs@#1#2#3#4#5{%
\sa{32}{#1}\sa{33}{#2}\sa{20}{#3}\sa{21}{#4}\sa{22}{#5}%
\WideBottleSetArgs@@}
\def\WideBottleSetArgs@@#1#2#3#4#5{%
\sa{23}{#1}\sa{10}{#2}\sa{11}{#3}\sa{12}{#4}\sa{13}{#5}%
\WideBottleSetArgs@@@}
\def\WideBottleSetArgs@@@#1#2#3#4{%
\sa{00}{#1}\sa{01}{#2}\sa{02}{#3}\sa{03}{#4}%
}
\makeatother
%R local WideBottle = 7/ !ga{33} \
%R | !ga{32} !ga{23} |
%R | !ga{22} !ga{13} |
%R | !ga{21} !ga{12} !ga{03}|
%R |!ga{20} !ga{11} !ga{02} |
%R | !ga{10} !ga{01} |
%R \ !ga{00} /
%R WideBottle:tomp({zdef="WideBottle", scale="7pt", meta="s"}):addcells():output()
%R WideBottle:tomp({zdef="WideBottleMed", scale="10pt", meta=""}):addcells():output()
\pu
\def\wibo #1{{ \WideBottleSetArgs{#1} \zha{WideBottle} }}
\def\pwibo#1{{\left( \WideBottleSetArgs{#1} \zha{WideBottle} \right)}}
\def\wiBo #1{{ \WideBottleSetArgs{#1} \zha{WideBottleMed} }}
\def\pwiBo#1{{\left( \WideBottleSetArgs{#1} \zha{WideBottleMed} \right)}}
% «SlantedHouse» (to ".SlantedHouse")
% (find-LATEX "2021groth-tops-defs.tex" "SlantedHouse")
%
%L SlantedHouse_ts = TCGSpec.new("32; 32,")
%L SlantedHouse_td_0 = TCGDims {h=15, v=8, q=15, crh=3.5, crv=7, qrh=5}
%L SlantedHouse_td_2 = TCGDims {h=65, v=50, q=15, crh=20, crv=15, qrh=5}
%L SlantedHouse_tq = TCGQ.newdsoa(SlantedHouse_td_0, SlantedHouse_ts,
%L {tdef="SlantedHouseSmall", meta="1pt s"},
%L "h ap")
%L SlantedHouse_tq:LRputs("!ga{L1} !ga{L2} !ga{L3}", "!ga{R1} !ga{R2}"):output()
%L
%L SlantedHouse_tq = TCGQ.newdsoa(SlantedHouse_td_2, SlantedHouse_ts,
%L {tdef="SlantedHouseBig", meta="1pt p"},
%L "h v ap")
%L SlantedHouse_tq:LRputs("!ga{L1} !ga{L2} !ga{L3}", "!ga{R1} !ga{R2}"):output()
%
\pu
%
\def\SlantedHouseSetargs#1#2#3#4#5{
\sa{L3}{#1}%
\sa{L2}{#2}\sa{R2}{#3}%
\sa{L1}{#4}\sa{R1}{#5}}
%
\def\SlantedHouse#1#2#3#4#5{{%
\SlantedHouseSetargs{#1}{#2}{#3}{#4}{#5}
\tcg{SlantedHouseSmall}}}
%
\def\SlantedHouseBig#1#2#3#4#5{{%
\SlantedHouseSetargs{#1}{#2}{#3}{#4}{#5}
\tcg{SlantedHouseBig}}}
%
\def\bsh#1#2#3#4#5{\left[ \SlantedHouse#1#2#3#4#5 \right]}
\def\bsht{\bsh01234}
% «ArtDecoN» (to ".ArtDecoN")
% (find-LATEX "2021groth-tops-defs.tex" "ArtDecoN")
%L ArtDecoN_ts = TCGSpec.new("33; 32,")
%L ArtDecoN_td_0 = TCGDims {h=15, v=8, q=15, crh=3.5, crv=7, qrh=5}
%L ArtDecoN_td_1 = TCGDims {h=25, v=22, q=15, crh=7.5, crv=7, qrh=5}
%L ArtDecoN_td_2 = TCGDims {h=65, v=50, q=15, crh=20, crv=15, qrh=5}
%L ArtDecoN_td_3 = TCGDims {h=85, v=70, q=15, crh=30, crv=30, qrh=5}
%L ArtDecoN_tq = TCGQ.newdsoa(ArtDecoN_td_0, ArtDecoN_ts,
%L {tdef="ArtDecoNSmall", meta="1pt s"},
%L "h ap")
%L ArtDecoN_tq:LRputs("!ga{L1} !ga{L2} !ga{L3}", "!ga{R1} !ga{R2} !ga{R3}"):output()
%L
%L ArtDecoN_tq = TCGQ.newdsoa(ArtDecoN_td_1, ArtDecoN_ts,
%L {tdef="ArtDecoNMed", meta="1pt s"},
%L "h v ap")
%L ArtDecoN_tq:LRputs("!ga{L1} !ga{L2} !ga{L3}", "!ga{R1} !ga{R2} !ga{R3}"):output()
%L
%L ArtDecoN_tq = TCGQ.newdsoa(ArtDecoN_td_2, ArtDecoN_ts,
%L {tdef="ArtDecoNBig", meta="1pt"},
%L "h v ap")
%L ArtDecoN_tq:LRputs("!ga{L1} !ga{L2} !ga{L3}", "!ga{R1} !ga{R2} !ga{R3}"):output()
%L
%L ArtDecoN_tq = TCGQ.newdsoa(ArtDecoN_td_3, ArtDecoN_ts,
%L {tdef="ArtDecoNBigg", meta="1pt"},
%L "h v ap")
%L ArtDecoN_tq:LRputs("!ga{L1} !ga{L2} !ga{L3}", "!ga{R1} !ga{R2} !ga{R3}"):output()
%
\pu
%
\def\ArtDecoNSetargs#1#2#3#4#5#6{
\sa{L3}{#1}\sa{R3}{#2}%
\sa{L2}{#3}\sa{R2}{#4}%
\sa{L1}{#5}\sa{R1}{#6}}
%
\def\ArtDecoN#1#2#3#4#5#6{{%
\ArtDecoNSetargs{#1}{#2}{#3}{#4}{#5}{#6}
\tcg{ArtDecoNSmall}}}
%
\def\ArtDecoNMed#1#2#3#4#5#6{{%
\ArtDecoNSetargs{#1}{#2}{#3}{#4}{#5}{#6}
\tcg{ArtDecoNMed}}}
%
\def\ArtDecoNBig#1#2#3#4#5#6{{%
\ArtDecoNSetargs{#1}{#2}{#3}{#4}{#5}{#6}
\tcg{ArtDecoNBig}}}
%
\def\ArtDecoNBigg#1#2#3#4#5#6{{%
\ArtDecoNSetargs{#1}{#2}{#3}{#4}{#5}{#6}
\tcg{ArtDecoNBigg}}}
%
\def\adn #1#2#3#4#5#6{ \ArtDecoN{#1}{#2}{#3}{#4}{#5}{#6} }
\def\padn#1#2#3#4#5#6{\left( \ArtDecoN{#1}{#2}{#3}{#4}{#5}{#6} \right)}
\def\badn#1#2#3#4#5#6{\left[ \ArtDecoN{#1}{#2}{#3}{#4}{#5}{#6} \right]}
\def\padnmed #1#2#3#4#5#6{\left( \ArtDecoNMed {#1}{#2}{#3}{#4}{#5}{#6} \right)}
\def\padnbig #1#2#3#4#5#6{\left( \ArtDecoNBig {#1}{#2}{#3}{#4}{#5}{#6} \right)}
\def\padnbigg#1#2#3#4#5#6{\left( \ArtDecoNBigg{#1}{#2}{#3}{#4}{#5}{#6} \right)}
% ----------------------------------------
{\setlength{\parindent}{0em}
\footnotesize
Notes on Bert Lindenhovius's
``Grothendieck topologies on posets''
\url{https://arxiv.org/abs/1405.4408v2}
\url{https://arxiv.org/abs/1405.4408v2.pdf}
\ssk
These notes are at:
\url{http://angg.twu.net/LATEX/2020lindenhovius.pdf}
}
% (find-books "__cats/__cats.el" "lindenhovius-gtop")
%D diagram ??
%D 2Dx 100 +35 +60 +80
%D 2D 100 A0 B0 C0 D0
%D 2D | | | |
%D 2D +30 A1 B1 C1 D1
%D 2D | | | |
%D 2D +30 A2 B2 C2 D2
%D 2D
%D ren A0 A1 A2 ==> 3 2 1
%D ren B0 B1 B2 ==> {↓}3=\{3,2,1\} {↓}2=\{2,1\} {↓}1=\{1\}
%D ren C0 ==> \calD({↓}3)=\csm{\{3,2,1\},\\\{2,1\},\\\{1\},\\∅}
%D ren C1 ==> \calD({↓}2)=\csm{\{2,1\},\\\{1\},\\∅}
%D ren C2 ==> \calD({↓}1)=\csm{\{1\},\\∅}
%D ren D0 ==> \calF({↓}3)=\Ftop
%D ren D1 ==> \calF({↓}3)=\Fmid
%D ren D2 ==> \calF({↓}1)=\Fbot
%D
%D (( A2 A1 -> A1 A0 ->
%D B0 place B1 place B2 place
%D C0 place C1 place C2 place
%D D0 place D1 place D2 place
%D ))
%D enddiagram
%D
$$\pu
\def\Ftop{\csm{\{\{3,2,1\}\},\\
\{\{3,2,1\},\{2,1\}\},\\
\{\{3,2,1\},\{2,1\},\{1\}\},\\
\{\{3,2,1\},\{2,1\},\{1\},∅\}}}
\def\Fmid{\csm{\{\{2,1\}\},\\
\{\{2,1\},\{1\}\},\\
\{\{2,1\},\{1\},∅\}}}
\def\Fbot{\csm{\{\{1\}\},\\
\{\{1\},∅\}}}
\diag{??}
$$
%D diagram ??-2
%D 2Dx 100 +35 +60 +80
%D 2D 100 A0 B0 C0 D0
%D 2D | | | |
%D 2D +30 A1 B1 C1 D1
%D 2D | | | |
%D 2D +30 A2 B2 C2 D2
%D 2D
%D ren A0 A1 A2 ==> 3 2 1
%D ren B0 B1 B2 ==> {↓}3=\{3,2,1\} {↓}2=\{2,1\} {↓}1=\{1\}
%D ren C0 ==> \calD({↓}3)=\csm{{↓}3,\\{↓}2,\\{↓}1,\\∅}
%D ren C1 ==> \calD({↓}3)=\csm{{↓}2,\\{↓}1,\\∅}
%D ren C2 ==> \calD({↓}3)=\csm{{↓}1,\\∅}
%D ren D0 ==> \calF(\calD({↓}3))=\Ftop
%D ren D1 ==> \calF(\calD({↓}3))=\Fmid
%D ren D2 ==> \calF(\calD({↓}1))=\Fbot
%D
%D (( A2 A1 -> A1 A0 ->
%D # B0 place B1 place B2 place
%D C0 place C1 place C2 place
%D D0 place D1 place D2 place
%D ))
%D enddiagram
%D
$$\pu
\def\Ftop{\csm{{↑}{↓}3,\\
{↑}{↓}2,\\
{↑}{↓}1,\\
{↑}∅}}
\def\Fmid{\csm{{↑}{↓}2,\\
{↑}{↓}1,\\
{↑}∅}}
\def\Fbot{\csm{{↑}{↓}1,\\
{↑}∅}}
\diag{??-2}
$$
\newpage
% 3. For $\calY = \cmat{\;\;\;\;\;\;▁3,\\1▁,▁1}$ we get:
%
% %L ArtDecoNQ_ts = TCGSpec.new("33; 32, ", ".??",".?.")
% %L ArtDecoNQ_ts:mp({zdef="ArtDecoNQ", scale="12pt", meta=""}):addlrs():output()
% \pu
% %
% \def\mygrotop{{
% \padnbig
% {\badn?·??11} {\badn·1·?·1}
% {\badn··?·1·} {\badn···?·1}
% {\badn····1·} {\badn·····1}
% }}
% \def\mygrotopz{{
% \padnbigg
% {\wibo{{32}· · {21}{22}· · {11}{12}· · · · · }}
% {\wibo{· · · · · · · · · · · {01}{02}{03}}}
% {\wibo{· · {20}· · · {10}· · · · · · · }}
% {\wibo{· · · · · · · · · · · {01}{02}· }}
% {\wibo{· · · · · · {10}· · · · · · · }}
% {\wibo{· · · · · · · · · · · {01}· · }}
% }}
% \def\mysubzha {\wiBo{{32}{33} {20}··· ···· {00}··{03}}}
% \def\mynucleus {\left( \zha{ArtDecoNQ} \right)_{(·)^*}}
% \def\mycongruence{\left( \zha{ArtDecoNQ} \right)_{(∼)}}
% \def\mysetofsieves{\cmat{\;\;\;\;\;\;▁3,\\1▁,▁1}}
% %
% $$\setlength{\arraycolsep}{0pt}
% \begin{array}{ccc}
% \mynucleus && \mysubzha \\
% &\mysetofsieves& \\
% \mycongruence && \scalebox{0.8}{$\mygrotop$} \\
% \end{array}
% $$
% (lindp 64 "B.25")
% (lind "B.25")
% (find-grtopsonposetspage 64 "Proposition B.25.")
% (find-grtopsonposetstext 64 "Proposition B.25.")
$$
\begin{array}{crll}
(\calY↦(·)^*) & \calS^* &= \calY\to \calS \\
(\calY↦H') & H' &= \{\calS\in H:\calS=(\calY\to\calS)\}=\{(\calY\to\calS):\calS∈H\} \\
(\calY↦∼) & ∼ &= \{(\calR,\calS)∈H^2:\calR∩\calY=\calS∩\calY\} \\
(\calY↦J) & J(u) &= \{\calS\inΩ(u):u\in (\calY\to \calS)\} \\
\end{array}
$$
and we will define some operations, with names like $(J \mapsto
\calY)$ and $(\calY \mapsto)$, that ``convert'' a $J$ to a $\calY$ and
vice-versa. We will define all these conversions first, then get some
visual intuition about how they work, and only then discuss which
composites of them are identities.
This section is about how to understand the ``essence'' of some
sections of \cite{Lindenhovius} from some examples. The precise
meaning of this ``essence'' will be discussed at the end.
% (find-grtopsonposetspage 48 "B Grothendieck topologies and Locale Theory")
\newpage
% «double-negation-old» (to ".double-negation-old")
% (grcp 28 "double-negation")
% (grc "double-negation")
{\bf The double negation topology:}
%L ArtDecoNQ_ts = TCGSpec.new("33; 32, ", ".??",".??")
%L ArtDecoNQ_ts = TCGSpec.new("33; 32, ", ".??",".?.")
%L ArtDecoNQ_ts:mp({zdef="WB_notnot", scale="12pt", meta=""}):addlrs():output()
\pu
%L ArtDecoNQ_ts = TCGSpec.new("33; 32, ", ".??",".??")
%L ArtDecoNQ_ts:mp({zdef="ArtDecoNQ", scale="12pt", meta=""}):addlrs():output()
\pu
\def\mygrotop{{
\padnbig
{\badn?·??11} {\badn·?·?·1}
{\badn··?·1·} {\badn···?·1}
{\badn····1·} {\badn·····1}
}}
\def\mygrotopz{{
\padnbigg
{\wibo{{32}· · {21}{22}· · {11}{12}· · · · · }}
{\wibo{· · · · · · · · · · · {01}{02}{03}}}
{\wibo{· · {20}· · · {10}· · · · · · · }}
{\wibo{· · · · · · · · · · · {01}{02}· }}
{\wibo{· · · · · · {10}· · · · · · · }}
{\wibo{· · · · · · · · · · · {01}· · }}
}}
\def\mysubzha {\wiBo{·{33} {20}··· ···· {00}··{03}}}
\def\mynucleus {\left( \zha{ArtDecoNQ} \right)_{(·)^*}}
\def\mycongruence{\left( \zha{ArtDecoNQ} \right)_{(∼)}}
$$\begin{array}{ccc}
\mynucleus && \mysubzha \\
\\
&\{1▁,▁1\}& \\
\\
\mycongruence && \mygrotop \\
\end{array}
$$
$$\scalebox{0.8}{$\mygrotopz$}
=\mygrotop
$$
From \cite{Lindenhovius}, proposition B.8, page 51:
%
% (find-grtopsonposetspage 51 "Proposition B.8.")
% (find-grtopsonposetstext 51 "Proposition B.8.")
% (lindp 51 "B.8")
% (lind "B.8")
%
%D diagram B.8
%D 2Dx 100 +45 +35 +10
%D 2D 100 A0 B0 C0 C1 D0
%D 2D || || | |
%D 2D +20 A1 B1 C2 C3 D1
%D 2D
%D 2D +15 E0 F0 F1 G0
%D 2D || | |
%D 2D +20 E1 F2 F3 G1
%D 2D
%D ren A0 A1 ==> \NucDP \GP
%D ren B0 B1 ==> j:\DP→\DP J∈\GP
%D ren C0 C1 C2 C3 ==> j j_J J_j J
%D ren E0 E1 ==> (·)^*:H→H J⊆Ω
%D ren F0 F1 F2 F3 ==> (·)^* (·)^* J J
%D
%D (( A0 A1 -> sl_
%D A0 A1 <- sl^
%D B0 B1 |-> sl_
%D B0 B1 <-| sl^
%D C0 C2 |->
%D C1 C3 <-|
%D E0 E1 |-> sl_
%D E0 E1 <-| sl^
%D F0 F2 |->
%D F1 F3 <-|
%D newnode: D0 at: @C1+v(60,0)
%D newnode: D1 at: @C3+v(60,0)
%D D0 .TeX= j_J(U):=\setofst{p∈\bfP}{U∩{↓}p∈J(p)} place
%D D1 .TeX= J_j(p):=\setofst{S∈\Ddp}{p∈j(S)} place
%D newnode: G0 at: @F1+v(60,0)
%D newnode: G1 at: @F3+v(60,0)
%D G0 .TeX= \calS^*:=\setofst{u∈H}{\calS∩{↓}u∈J(u)} place
%D G1 .TeX= J(u):=\setofst{\calS∈Ω(u)}{u∈\calS^*} place
%D ))
%D enddiagram
%D
$$\pu
\diag{B.8}
$$
From \cite{Lindenhovius}, proposition B.12, page 55:
%
% (find-grtopsonposetspage 55 "Proposition B.12")
% (find-grtopsonposetstext 55 "Proposition B.12")
%
%D diagram B.12
%D 2Dx 100 +45 +35 +10
%D 2D 100 A0 B0 C0 C1 D0
%D 2D || || | |
%D 2D +20 A1 B1 C2 C3 D1
%D 2D
%D 2D +15 E0 F0 F1 G0
%D 2D || | |
%D 2D +20 E1 F2 F3 G1
%D 2D
%D ren A0 A1 ==> \NucDP \Sub(\DP)^\op
%D ren B0 B1 ==> j:\DP→\DP M⊆\DP
%D ren C0 C1 C2 C3 ==> j j_M M_j J
%D ren E0 E1 ==> (·)^*:H→H H'⊆H
%D ren F0 F1 F2 F3 ==> (·)^* (·)^* H' H'
%D
%D (( A0 A1 -> sl_
%D A0 A1 <- sl^
%D B0 B1 |-> sl_
%D B0 B1 <-| sl^
%D C0 C2 |->
%D C1 C3 <-|
%D E0 E1 |-> sl_
%D E0 E1 <-| sl^
%D F0 F2 |->
%D F1 F3 <-|
%D newnode: D0 at: @C1+v(60,0)
%D newnode: D1 at: @C3+v(60,0)
%D D0 .TeX= j_M(a):=\bigwedge\setofst{m∈M}{a≤m} place
%D D1 .TeX= M_j:=\setofst{x∈L}{j(x)=x} place
%D newnode: G0 at: @F1+v(60,0)
%D newnode: G1 at: @F3+v(60,0)
%D G0 .TeX= \calR^*:=\bigwedge\setofst{\calS∈H'}{\calR≤\calS} place
%D G1 .TeX= H':=\setofst{\calR∈H}{\calR^*=\calR} place
%D ))
%D enddiagram
%D
$$\pu
\diag{B.12}
$$
From \cite{Lindenhovius}, proposition B.23, page 63:
%
% (find-grtopsonposetspage 63 "Proposition B.23.")
% (find-grtopsonposetstext 63 "Proposition B.23.")
%
%D diagram B.23
%D 2Dx 100 +45 +35 +10
%D 2D 100 A0 B0 C0 C1 D0
%D 2D || || | |
%D 2D +20 A1 B1 C2 C3 D1
%D 2D
%D 2D +15 E0 F0 F1 G0
%D 2D || | |
%D 2D +20 E1 F2 F3 G1
%D 2D
%D ren A0 A1 ==> \NucDP \ConDP
%D ren B0 B1 ==> j:\DP→\DP θ⊆\DP^2
%D ren C0 C1 C2 C3 ==> j j_θ θ_j θ
%D ren E0 E1 ==> (·)^*:H→H ∼\;⊆H×H
%D ren F0 F1 F2 F3 ==> (·)^* (·)^* ∼ ∼
%D
%D (( A0 A1 -> sl_
%D A0 A1 <- sl^
%D B0 B1 |-> sl_
%D B0 B1 <-| sl^
%D C0 C2 |->
%D C1 C3 <-|
%D E0 E1 |-> sl_
%D E0 E1 <-| sl^
%D F0 F2 |->
%D F1 F3 <-|
%D newnode: D0 at: @C1+v(60,0)
%D newnode: D1 at: @C3+v(60,0)
%D D0 .TeX= j_θ(a):=\bigvee\setofst{b∈\DP}{aθb} place
%D D1 .TeX= θ_j:=\setofst{(a,b)∈\DP^2}{j(a)=j(b)} place
%D newnode: G0 at: @F1+v(60,0)
%D newnode: G1 at: @F3+v(60,0)
%D G0 .TeX= \calS^*:=\bigvee\setofst{\calR∈H}{\calR∼\calS} place
%D G1 .TeX= ∼\;:=\setofst{(\calR,\calS)∈H^2}{\calR^*=\calS^*} place
%D ))
%D enddiagram
%D
$$\pu
\diag{B.23}
$$
\newpage
From \cite{Lindenhovius}, theorem B.25, page 64...
%
% (lindp 64 "B.25")
% (lind "B.25")
% (find-grtopsonposetspage 64 "Proposition B.25.")
% (find-grtopsonposetstext 64 "Proposition B.25.")
%
%D diagram ??
%D 2Dx 100 +60 +30 +25 +20 +25
%D 2D 100 A0 - A1 B0 - B1 C0 - C1
%D 2D | | | | | |
%D 2D +25 A2 - A3 B2 - B3 C2 - C3
%D 2D
%D ren A0 A1 A2 A3 ==> \NucDP \Sub(\DP)^\op \ConDP \GP
%D ren B0 B1 B2 B3 ==> j M θ J
%D ren C0 C1 C2 C3 ==> (·)^* H' ∼ J
%D
%D (( A0 A1 -> .plabel= a j↦\calM_j
%D A0 A2 -> .plabel= l j↦θ_j
%D A0 A3 -> .plabel= m j↦J_j
%D A1 A3 -> .plabel= r \calM↦J_\calM
%D A2 A3 -> .plabel= b θ↦J_θ
%D
%D B0 B1 |->
%D B0 B2 |->
%D B0 B3 |->
%D B1 B3 |->
%D B2 B3 |->
%D
%D C0 C1 |-> sl^
%D C0 C1 <-| sl_
%D C0 C2 |-> sl^
%D C0 C2 <-| sl_
%D C0 C3 |-> sl^
%D C0 C3 <-| sl_
%D C1 C3 |-> sl^
%D C1 C3 <-| sl_
%D C2 C3 |-> sl^
%D C2 C3 <-| sl_
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
\def\M{\mathcal{M}}
\def\D{\mathcal{D}}
\def\G{\mathcal{G}}
\def\P{\mathbf{P}}
\def\down{{↓}}
$$
\begin{array}{rll}
J_\M(p) & = \{S\in\D(\down p):\forall M\in\M(S\subseteq M\implies p\in M)\}; & \M\in\Sub(\D(\P)) \\
J_j(p) & = \{S\in\D(\down p):p\in j(S)\}; & j\in\Nuc(\D(\P)) \\
J_\theta(p) & = \{S\in\D(\down p):S\theta\down p\}; & \theta\in\Con(\D(\P)) \\
j_J(A) & = \{p\in\P:A\cap\down p\in J(p)\}; &J\in\G(\P)\\
j_\theta(A) & = \bigcup\{B\in\D(\P):B\theta A\}; & \theta\in\Con(\D(\P))\\
j_\M(A) & = \bigcap\{A\in\M:A\subseteq M\}; & \M\in\Sub(\D(\P)) \\
\theta_j& = \ker j=\{(A,B)\in\D(\P)^2:j(A)=j(B)\}; & j\in\Nuc(\D(\P))\\
\theta_J & = \{(A,B)\in\D(\P)^2:\forall p\in\P(A\cap\down p\in J(p)\Longleftrightarrow B\cap\down p\in J(p))\}; &J\in\G(\P) \\
\M_j & = \{A\in\D(\P):j(A)=A\}=j[\D(\P)]; & j\in\Nuc(\D(\P))\\
\M_J & = \{A\in\D(\P):\forall p\in\P(A\cap\down p\in J(p)\implies p\in A)\}; &J\in\G(\P) \\
\end{array}
$$
From \cite{Lindenhovius}, theorem C.4, page 74...
%
$$
\begin{array}{crll}
(X↦J) & J_X(p) &= \{S\in\D(\down p):p\in X\to S\} \\
(\calY↦J) & J(u) &= \{\calS\inΩ(u):u\in (\calY\to \calS)\} \\
\\
(X↦j) & j_X(A) &= X\to A \\
(\calY↦(·)^*) & \calS^* &= \calY\to \calS \\
\\
(X↦θ) & \theta_X &= \ker i_X^{-1}= \{(A,B)\in\D(\P)^2:A\cap X=B\cap X\} \\
(\calY↦∼) & ∼ &= \{(\calR,\calS)∈H^2:\calR∩\calY=\calS∩\calY\} \\
\\
(X↦\M) & \M_X &= \{A\in\D(\P):A=X\to A\}=\{X\to A:A\in\D(\P)\} \\
(\calY↦H') & H' &= \{\calS\in H:\calS=(\calY\to\calS)\}=\{(\calY\to\calS):\calS∈H\} \\
\end{array}
$$
%
% (lindp 74 "C.4")
% (lind "C.4")
% (find-grtopsonposetspage 74 "Theorem C.4")
% (find-grtopsonposetstext 74 "Theorem C.4")
%
%D diagram ??
%D 2Dx 100 +30 +30 +30 +15 +15 +20 +15 +15
%D 2D 100 A0 ---- A1 B0 ---- B1 C0 ---- C1
%D 2D | \ / | | \ / | | \ / |
%D 2D +15 | A4 | | B4 | | C4 |
%D 2D | / \ | | / \ | | / \ |
%D 2D +15 A2 ---- A3 B2 ---- B3 C2 ---- C3
%D 2D
%D ren A0 A1 A2 A3 A4 ==> \NucDP \Sub(\DP)^\op \ConDP \GP \Pts(\bfP)^\op
%D ren B0 B1 B2 B3 B4 ==> j M θ J ?
%D ren C0 C1 C2 C3 C4 ==> (·)^* H' ∼ J \calY
%D
%D (( A0 A1 -> .plabel= a j↦\calM_j
%D A0 A2 -> .plabel= l j↦θ_j
%D # A0 A3 -> .plabel= m j↦J_j
%D A1 A3 -> .plabel= r \calM↦J_\calM
%D A2 A3 -> .plabel= b θ↦J_θ
%D A4 A0 ->
%D A4 A1 ->
%D A4 A2 ->
%D A4 A3 ->
%D
%D B0 B1 |->
%D B0 B2 |->
%D # B0 B3 |->
%D B1 B3 |->
%D B2 B3 |->
%D B4 B0 |->
%D B4 B1 |->
%D B4 B2 |->
%D B4 B3 |->
%D
%D C0 C1 |-> sl^
%D C0 C1 <-| sl_
%D C0 C2 |-> sl^
%D C0 C2 <-| sl_
%D # C0 C3 |-> sl^
%D # C0 C3 <-| sl_
%D C1 C3 |-> sl^
%D C1 C3 <-| sl_
%D C2 C3 |-> sl^
%D C2 C3 <-| sl_
%D C4 C0 |->
%D C4 C1 |->
%D C4 C2 |->
%D C4 C3 |->
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
\newpage
% (fooi "\\cap" "∩" "\\down" "{↓}" "\\subseteq" "⊆" "\\in" "∈" "\\forall" "∀" "\\theta" "θ")
$$
\begin{array}{crll}
(\M↦J) & J_\M(p) & = \{S∈\D({↓} p):∀ M∈\M(S⊆ M\implies p∈ M)\} \\
(H'↦J) & J(u) & = \{\calS∈Ω(u):∀\calT∈H'.\;(\calS⊆\calT\implies u∈\calT)\} \\
\\
(j↦J) & J_j(p) & = \{S∈\D({↓} p):p∈ j(S)\} \\
((·)^*↦J) & J(u) & = \{\calS∈Ω(u):u∈\calS^*\} \\
\\
(θ↦J) & J_θ(p) & = \{S∈\D({↓} p):Sθ{↓} p\} \\
(∼↦J) & J(u) & = \{\calS∈Ω(u):\calS∼{↓}u\} \\
\\
(J↦j) & j_J(A) & = \{p∈\P:A∩{↓} p∈ J(p)\} \\
(J↦(·)^*) & \calS^* & = \{u∈D:\calS∩{↓}u∈J(u)\} \\
\\
(θ↦j) & j_θ(A) & = \bigcup\{B∈\D(\P):Bθ A\}\\
(∼↦(·)^*) & \calS^* & = \bigcup\{\calR∈H:\calR∼\calS\}\\
\\
(\M↦j) & j_\M(A) & = \bigcap\{A∈\M:A⊆ M\} \\
(H'↦(·)^*) & \calS^* & = \bigcap\{\calT∈H':\calS⊆\calT\} \\
\\
(j↦θ) & θ_j & = \ker j=\{(A,B)∈\D(\P)^2:j(A)=j(B)\}\\
((·)^*↦∼) & ∼ & = \ker j=\{(\calR,\calS)∈H^2:\calR^*=\calS^*\}\\
\\
(J↦θ) & θ_J & = \{(A,B)∈\D(\P)^2:∀ p∈\P(A∩{↓} p∈ J(p) \Leftrightarrow B∩{↓} p∈ J(p))\} \\
(J↦∼) & ∼ & = \{(\calR,\calS)∈H^2:∀ u∈D.\;(\calR∩{↓}u∈J(u) \Leftrightarrow \calS∩{↓}u∈ J(u))\} \\
\\
(j↦\M) & \M_j & = \{A∈\D(\P):j(A)=A\}=j[\D(\P)] \\
((·)^*↦H') & H' & = \{\calS∈H:\calS^*=\calS\}=H^* \\
\\
(J↦\M) & \M_J & = \{A∈\D(\P):∀ p∈\P(A∩{↓} p∈ J(p) ⇒ p∈ A)\} \\
(J↦H') & H' & = \{\calS∈H:∀ u∈D.\; (\calS∩{↓}u∈J(u) ⇒ u∈\calS)\} \\
\end{array}
$$
\newpage
%D diagram ??
%D 2Dx 100 +15 +20 +15 +15 +20 +15 +15
%D 2D 100 B0 ---- B1 C0 ---- C1
%D 2D | \ | | \ / |
%D 2D +15 A4 | | | C4 |
%D 2D \ | \ | | / \ |
%D 2D +15 A3 B2 ---- B3 C2 ---- C3
%D 2D
%D ren A3 A4 ==> J \calY
%D ren B0 B1 B2 B3 ==> (·)^* H' ∼ J
%D ren C0 C1 C2 C3 C4 ==> (·)^* H' ∼ J \calY
%D
%D (( A4 A3 |-> sl^
%D A4 A3 <-| sl_
%D
%D B0 B1 |-> sl^
%D B0 B1 <-| sl_
%D B0 B2 |-> sl^
%D B0 B2 <-| sl_
%D B0 B3 |-> sl^
%D B0 B3 <-| sl_
%D B1 B3 |-> sl^
%D B1 B3 <-| sl_
%D B2 B3 |-> sl^
%D B2 B3 <-| sl_
%D
%D C0 C1 |-> sl^
%D C0 C1 <-| sl_
%D C0 C2 |-> sl^
%D C0 C2 <-| sl_
%D # C0 C3 |-> sl^
%D # C0 C3 <-| sl_
%D C1 C3 |-> sl^
%D C1 C3 <-| sl_
%D C2 C3 |-> sl^
%D C2 C3 <-| sl_
%D C4 C0 |->
%D C4 C1 |->
%D C4 C2 |->
%D C4 C3 |->
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
$$
\begin{array}{crll}
(\calY↦J) & J(u) &= \{\calS∈Ω(u):\calY∩{↓}u⊆\calS\} \\
(J↦\calY) & \calY &= \{u∈D:J(u)=\{{↓}u\}\} \\
\\
((·)^*↦H') & H' & = \{\calS∈H:\calS^*=\calS\}=H^* \\
(H'↦(·)^*) & \calS^* & = \bigcap\{\calT∈H':\calS⊆\calT\} \\
\\
((·)^*↦∼) & ∼ & = \{(\calR,\calS)∈H^2:\calR^*=\calS^*\}\\
(∼↦(·)^*) & \calS^* & = \bigcup\{\calR∈H:\calR∼\calS\}\\
\\
((·)^*↦J) & J(u) & = \{\calS∈Ω(u):u∈\calS^*\} \\
(J↦(·)^*) & \calS^* & = \{u∈D:\calS∩{↓}u∈J(u)\} \\
\\
(H'↦J) & J(u) & = \{\calS∈Ω(u):∀\calT∈H'.\;(\calS⊆\calT ⇒ u∈\calT)\} \\
(J↦H') & H' & = \{\calS∈H:∀ u∈D.\; (\calS∩{↓}u∈J(u) ⇒ u∈\calS)\} \\
\\
(∼↦J) & J(u) & = \{\calS∈Ω(u):\calS∼{↓}u\} \\
(J↦∼) & ∼ & = \{(\calR,\calS)∈H^2:∀ u∈D.\;(\calR∩{↓}u∈J(u) ↔ \calS∩{↓}u∈ J(u))\} \\
\\
(\calY↦(·)^*) & \calS^* &= \calY\to \calS \\
(\calY↦H') & H' &= \{\calS\in H:\calS=(\calY\to\calS)\}=\{(\calY\to\calS):\calS∈H\} \\
(\calY↦∼) & ∼ &= \{(\calR,\calS)∈H^2:\calR∩\calY=\calS∩\calY\} \\
(\calY↦J) & J(u) &= \{\calS\inΩ(u):u\in (\calY\to \calS)\} \\
\end{array}
$$
These are some other constructions that I am starting to translate...
$$\begin{array}{rcl}
(·)^* &:& \Downs(\Opens(B)) → \Downs(\Opens(B)) \\
(·)^* &:& \Downs(\Opens(X)) → \Downs(\Opens(X)) \\
Ω(U) &=& \Downs(\Opens(U)) \\
\Jcan(U) &=& \setofst{\calS∈Ω(U)}{\calS^*={↓}U} \\
J(U)(\calS) &=& \calS^* \\
\\
(·)^* &:& \Downs(X) → \Downs(X) \\
Ω(u) &=& \Downs({↓}u) \\
\Jcan(u) &=& \setofst{\calS∈Ω(u)}{\calS^*={↓}u} \\
J(u)(\calS) &=& \calS^* \\
\\
J_j(p) &:=& \setofst{S∈\calD({↓}p)}{p∈j(S)} \\
J(U) &:=& \setofst{\calS∈\Downs({↓}U)}{U∈\calS^*} \\
\\
j_J(U) &:=& \setofst{p∈𝐛P}{p∈j(S)} \\
\calS^* &:=& \setofst{V∈\Opens(B)}{\calS∩{↓}V∈J(V)} \\
\\
𝐛P &≡& B \\
\calD(𝐛P) &≡& \Opens(B) \\
\mathrm{Nuc}(\calD(𝐛P)) &≡& \setofst{ (·)^*: \Opens(B)→\Opens(B) }{ (·)^* \text{ is a J-operator}} \\
\calG(𝐛P) &≡& \setofst{ J⊆Ω_{\Set^{\Opens(B)^\op}} }{ J \text{ is a Gr.top.}} \\
\end{array}
$$
% \section{Mac Lane/Moerdijk}
%
% \cite[section V.1, page 38]{MacLaneMoerdijk}
%
% % (find-books "__cats/__cats.el" "maclane-moerdijk")
% % (find-maclanemoerdijkpage (+ 11 38) "Sieve on C =")
% % (find-maclanemoerdijkpage (+ 11 38) "Omega(C) =")
% % (find-maclanemoerdijkpage (+ 11 38) "t(C)")
% % (find-maclanemoerdijkpage (+ 11 110) "Definition 1. A Grothendieck Topology")
%
% $$\begin{array}{rcl}
% \text{Sieve on $C$} &=& \text{Subfunctor of $\Hom_\catC(-,C)$} \\
% Ω(C) &=& \setofst{S}{\text{$S$ is a sieve on $C$ in $\catC$}} \\
% t(C) &=& \setofst{h}{\cod(h) = C} \\
% \end{array}
% $$
%
% And if $g:C'→C$ is an arrow in $\catC$ then:
% %
% $$\begin{array}{rrcl}
% (-)·g: &Ω(C)& →& Ω(C')\\
% & S & ↦& S·g = \setofst{h}{g∘h∈S} \\
% \end{array}
% $$
%
% %D diagram ??
% %D 2Dx 100 +25 +30
% %D 2D 100 A0 - A1 C0
% %D 2D | | |
% %D 2D +20 A2 - A3 C1
% %D 2D
% %D 2D +20 B0 - B1
% %D 2D
% %D ren A0 A1 A2 A3 ==> C Ω(C) C' Ω(C')
% %D ren C0 C1 ==> S S·g
% %D ren B0 B1 ==> \catC^\op \Set
% %D
% %D (( A0 A1 |->
% %D A0 A2 <- .plabel= l g
% %D A1 A3 -> .plabel= r (-)·g
% %D A0 A3 harrownodes nil 20 nil |->
% %D A2 A3 |->
% %D newnode: B0' at: @B0+v(0,-8) .TeX= \catC place
% %D B0 B1 ->
% %D C0 C1 |->
% %D ))
% %D enddiagram
% %D
% $$\pu
% \diag{??}
% $$
%
\newpage
2021jun20:
%D diagram ??
%D 2Dx 100 +40
%D 2D 100 A0 - A1
%D 2D | /
%D 2D +40 A2 - A3
%D 2D
%D ren A0 A1 A2 A3 ==> 𝓨 \nuc J j
%D
%D (( A0 A1 |-> sl^ .plabel= a C.4.2
%D A0 A1 <--| sl_
%D A0 A2 |-> sl_ .plabel= l \sm{2.8,\\C.4.1}
%D A0 A2 <-| sl^ .plabel= r 2.9
%D A2 A1 <-> .plabel= m \sm{B.8,\\B.25}
%D A2 A3 <-->
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
Definition C.2:
%
% (lindp 64 "B.25")
% (linda "B.25")
%
$$\begin{array}{rcl}
X→Y &=& \bigcup \setofsc{A∈\DP}{A∩X⊆Y} \\
𝓨→𝓩 &=& \bigcup \setofst{𝓢∈H}{𝓢∩𝓨⊆𝓩} \\
&=& \bigcup \setofst{𝓢∈H}{𝓢 ⊆ 𝓨 →_M 𝓩} \\
&=& \bigcup \setofst{𝓢∈H}{𝓢 ⊆ \Int(𝓨 →_M 𝓩)} \\
&=& \Int(𝓨 →_M 𝓩) \\
\end{array}
$$
2.8, C.4.1:
%
% (lindp 11 "2.8")
% (linda "2.8")
% (lindp 74 "C.4")
% (linda "C.4")
%
$$\begin{array}{lcr}
J_X(p) &=& \setofsc{S∈\Ddp}{X∩{↓}p⊆S} \\
J_X &=& λp∈𝐏. \; \setofsc{S∈\Ddp}{X∩{↓}p⊆S} \\
(X↦J)(X) &=& λp∈𝐏. \; \setofsc{S∈\Ddp}{X∩{↓}p⊆S} \\
(X↦J) &=& λX∈\Pts(𝐏). \; λp∈𝐏. \; \setofsc{S∈\Ddp}{X∩{↓}p⊆S} \\
[5pt]
J_X(p) &=& \setofsc{S∈\Ddp}{p∈X→S} \\
J_X &=& λp∈𝐏. \; \setofsc{S∈\Ddp}{p∈X→S} \\
(X↦J)(X) &=& λp∈𝐏. \; \setofsc{S∈\Ddp}{p∈X→S} \\
(X↦J) &=& λX∈\Pts(𝐏). \; λp∈𝐏. \; \setofsc{S∈\Ddp}{p∈X→S} \\
[5pt]
(𝓨↦J) &=& λ𝓨∈\Pts(𝐃_0). \; λu∈𝐃_0. \; \setofst{𝓢∈Ω(u)}{u∈𝓨→𝓢} \\
&=& λ𝓨∈\Pts(𝐃_0). \; λu∈𝐃_0. \; \setofst{𝓢∈Ω(u)}{{↓}u⊆𝓨→𝓢} \\
&=& λ𝓨∈\Pts(𝐃_0). \; λu∈𝐃_0. \; \setofst{𝓢∈Ω(u)}{{↓}u∩𝓨⊆𝓢} \\
&=& λ𝓨∈\Pts(𝐃_0). \; λu∈𝐃_0. \; \setofst{𝓢∈Ω(u)}{𝓨∩{↓}u⊆𝓢} \\
\end{array}
$$
% 2.8:
% %
% % (lindp 11 "2.8")
% % (linda "2.8")
% %
% $$\begin{array}{lcr}
% J_X(p) &=& \setofsc{S∈\Ddp}{X∩{↓}p⊆S} \\
% J_X &=& λp∈𝐏. \; \setofsc{S∈\Ddp}{X∩{↓}p⊆S} \\
% (X↦J)(X) &=& λp∈𝐏. \; \setofsc{S∈\Ddp}{X∩{↓}p⊆S} \\
% (X↦J) &=& λX∈\Pts(𝐏).\; λp∈𝐏. \; \setofsc{S∈\Ddp}{X∩{↓}p⊆S} \\
% (𝓨↦J) &=& λ𝓨∈\Pts(𝐃_0).\; λu∈𝐃_0. \; \setofst{𝓢∈Ω(u)}{𝓨∩{↓}u⊆𝓢} \\
% \end{array}
% $$
2.9:
%
% (lindp 12 "2.9")
% (linda "2.9")
%
$$\begin{array}{lcr}
X_J &=& \setofsc{p∈𝐏}{J(p)=\{{↓}p\}} \\
(J↦X)(J) &=& \setofsc{p∈𝐏}{J(p)=\{{↓}p\}} \\
(J↦X) &=& λJ∈\G(𝐏). \; \setofsc{p∈𝐏}{J(p)=\{{↓}p\}} \\
(J↦𝓨) &=& λJ∈\GrTops(𝐃). \; \setofst{u∈𝐃_0}{J(u)=\{{↓}u\}} \\
\end{array}
$$
B.8, B.25:
%
% (lindp 64 "B.25")
% (linda "B.25")
%
$$\begin{array}{lcr}
J_j(p) &=& \setofsc{S∈\Ddp}{p∈j(S)} \\
J_j &=& λp∈𝐏. \; \setofsc{S∈\Ddp}{p∈j(S)} \\
(j↦J)(j) &=& λp∈𝐏. \; \setofsc{S∈\Ddp}{p∈j(S)} \\
(j↦J) &=& λj∈\Nuc(\DP). \; λp∈𝐏. \; \setofsc{S∈\Ddp}{p∈j(S)} \\[5pt]
(\nuc↦J) &=& λ\nuc∈\Nucs(H). \; λu∈𝐃_0. \; \setofst{𝓢∈Ω(u)}{u∈𝓢^*} \\
\\
j_J(A) &=& \setofsc{p∈𝐏}{A∩{↓}p∈J(p)} \\
j_J &=& λA∈\DP. \; \setofsc{p∈𝐏}{A∩{↓}p∈J(p)} \\
(J↦j)(J) &=& λA∈\DP. \; \setofsc{p∈𝐏}{A∩{↓}p∈J(p)} \\
(J↦j) &=& λJ∈\G(𝐏). \; λA∈\DP. \; \setofsc{p∈𝐏}{A∩{↓}p∈J(p)} \\[5pt]
(J↦\nuc) &=& λJ∈\GrTops(𝐃). \; λ𝓢∈H. \; \setofst{u∈𝐃_0}{𝓢∩{↓}u∈J(u)} \\
\end{array}
$$
C.4.2:
%
% (lindp 74 "C.4")
% (linda "C.4")
%
$$\begin{array}{lcr}
j_X(A) &=& X→A \\
j_X &=& λA∈\DP. \; X→A \\
(X↦j)(X) &=& λA∈\DP. \; X→A \\
(X↦j) &=& λX∈\Pts(𝐏). \; λA∈\DP. \; X→A \\[5pt]
(𝓨↦\nuc) &=& λ𝓨∈\Pts(𝐃_0). \; λ𝓢∈H. \; 𝓨→𝓢 \\
\end{array}
$$
\newpage
% https://mail.google.com/mail/ca/u/0/#sent/QgrcJHrtvWmlxhvCBggGFXMszBggGkmQmdv
My hypothesis about C1:
$$\begin{array}{lcr}
X_j &=& \setofsc{p∈𝐏}{j({↓}p) ≠ j({↓}p∖\{p\})} \\
(j↦X)(j) &=& \setofsc{p∈𝐏}{j({↓}p) ≠ j({↓}p∖\{p\})} \\
(j↦X) &=& λj∈\Nuc(\DP). \; \setofsc{p∈𝐏}{j({↓}p) ≠ j({↓}p∖\{p\})} \\
[5pt]
(\nuc↦𝓨) &=& λ\nuc∈\Nucs(H). \; \setofst{u∈𝐃_0}{({↓}u)^* ≠ ({↓}u∖\{u\})^*} \\
\end{array}
$$
\bsk
Trying to decypher the real C1:
C.1, p.70:
\def\iYm{{i_Y^{-1}}}
\def\iff{\text{iff}}
$$\begin{array}{rcr}
X_f & = & \setofsc{p∈𝐏}{f({↓}p) ≠ f({↓}p∖\{p\})} \\
[5pt]
p∈X_\iYm & \iff & \iYm({↓}p) ≠ \iYm({↓}p∖\{p\}) \\
& \iff & Y∩{↓}p ≠ Y∩({↓}p∖\{p\}) \\
X_\iYm & = & \setofsc{p∈𝐏}{Y∩{↓}p ≠ Y∩({↓}p∖\{p\})} \\
\end{array}
$$
%D diagram C1a
%D 2Dx 100 +30 +30
%D 2D 100 A00 = A0 <-| A1
%D 2D
%D 2D +15 B0 <-- B1
%D 2D
%D 2D +15 C0 `-> C1
%D 2D
%D ren A00 A0 A1 ==> A∩Y \iYm(A) A
%D ren B0 B1 ==> \D(Y) \DP
%D ren C0 C1 ==> Y 𝐏
%D
%D (( A00 A0 = A0 A1 <-|
%D B0 B1 <- .plabel= a \iYm
%D C0 C1 `-> .plabel= a i_Y
%D ))
%D enddiagram
%D
$$\pu
\diag{C1a}
$$
%D diagram ??
%D 2Dx 100 +40
%D 2D 100 A0 A1
%D 2D
%D 2D +20 A2 A3
%D 2D
%D 2D +15 B0 B1
%D 2D
%D ren A0 A1 ==> X_f [f]_E
%D ren A2 A3 ==> Y [\iYm]_E
%D ren B0 B1 ==> \Pts(𝐏)^\op \calE(\DP)
%D
%D (( A0 A1 <-|
%D A0 A2 -> A1 A3 ->
%D A2 A3 |->
%D B0 B1 <- sl^ .plabel= a F
%D B0 B1 <- sl_ .plabel= b G
%D
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
$$X_f = \setofsc{p∈𝐏}{f({↓}p) ≠ f({↓}p∖\{p\})}$$
\newpage
$$\begin{array}{rcl}
𝓨→𝓩 &=& \bigcup \setofst{𝓢∈H}{𝓢∩𝓨⊆𝓩} \\
&=& \bigcup \setofst{𝓢∈H}{𝓢 ⊆ 𝓨 →_M 𝓩} \\
&=& \bigcup \setofst{𝓢∈H}{𝓢 ⊆ \Int(𝓨 →_M 𝓩)} \\
&=& \Int(𝓨 →_M 𝓩) \\
%
[5pt]
%
(𝓨↦\nuc) &=& λ𝓨∈\Pts(𝐃_0). \; λ𝓢∈H. \; 𝓨→𝓢 \\
(\nuc↦𝓨) &=& λ\nuc∈\Nucs(H). \; \setofst{u∈𝐃_0}{({↓}u)^* ≠ ({↓}u∖\{u\})^*} \\
%
[5pt]
%
(𝓨↦J) &=& λ𝓨∈\Pts(𝐃_0). \; λu∈𝐃_0. \; \setofst{𝓢∈Ω(u)}{u∈𝓨→𝓢} \\
(J↦𝓨) &=& λJ∈\GrTops(𝐃). \; \setofst{u∈𝐃_0}{J(u)=\{{↓}u\}} \\
%
[5pt]
%
(\nuc↦J) &=& λ\nuc∈\Nucs(H). \; λu∈𝐃_0. \; \setofst{𝓢∈Ω(u)}{u∈𝓢^*} \\
(J↦\nuc) &=& λJ∈\GrTops(𝐃). \; λ𝓢∈H. \; \setofst{u∈𝐃_0}{𝓢∩{↓}u∈J(u)} \\
%
%[5pt]
\end{array}
$$
% (find-books "__cats/__cats.el" "lindenhovius-gtop")
\GenericWarning{Success:}{Success!!!} % Used by `M-x cv'
%\printbibliography
\end{document}
% __ __ _
% | \/ | __ _| | _____
% | |\/| |/ _` | |/ / _ \
% | | | | (_| | < __/
% |_| |_|\__,_|_|\_\___|
%
% <make>
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2020lindenhovius veryclean
make -f 2019.mk STEM=2020lindenhovius pdf
% Local Variables:
% coding: utf-8-unix
% ee-tla: "lin"
% End: