% (find-LATEX "2021groth-tops-children-slides.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2021groth-tops-children-slides.tex" :end))
% (defun C () (interactive) (find-LATEXSH "lualatex 2021groth-tops-children-slides.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2021groth-tops-children-slides.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2021groth-tops-children-slides.pdf"))
% (defun e () (interactive) (find-LATEX "2021groth-tops-children-slides.tex"))
% (defun l () (interactive) (find-LATEX "2021groth-tops-children.lua"))
% (defun u () (interactive) (find-latex-upload-links "2021groth-tops-children-slides"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (defun d0 () (interactive) (find-ebuffer "2021groth-tops-children-slides.pdf"))
%          (code-eec-LATEX "2021groth-tops-children-slides")
% (find-pdf-page   "~/LATEX/2021groth-tops-children-slides.pdf")
% (find-sh0 "cp -v  ~/LATEX/2021groth-tops-children-slides.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2021groth-tops-children-slides.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2021groth-tops-children-slides.pdf
%               file:///tmp/2021groth-tops-children-slides.pdf
%           file:///tmp/pen/2021groth-tops-children-slides.pdf
% http://angg.twu.net/LATEX/2021groth-tops-children-slides.pdf
% (find-LATEX "2019.mk")

% «.defs»		(to "defs")
%   «.defs-Bottle»	(to "defs-Bottle")
%   «.defs-GRO»		(to "defs-GRO")
% «.title»		(to "title")
% «.abstract»		(to "abstract")
% «.what-we-will-need»	(to "what-we-will-need")
% «.typical-letters»	(to "typical-letters")
% «.typical-values»	(to "typical-values")
% «.quotient-tops»	(to "quotient-tops")
% «.2CGs»		(to "2CGs")
% «.GRO»		(to "GRO")
% «.intervals-top»	(to "intervals-top")
% «.lindenhovius-filter»	(to "lindenhovius-filter")
% «.classifying-map-1»	(to "classifying-map-1")
% «.LittleN»		(to "LittleN")
% «.OLittleN»		(to "OLittleN")
% «.classifier-LittleN»	(to "classifier-LittleN")
% «.bijections-1»	(to "bijections-1")

\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")
\usepackage[a6paper, landscape,
            top=1.5cm, bottom=.25cm, left=1cm, right=1cm, includefoot
           ]{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")
%L dofile "2021groth-tops-children.lua"  -- (find-LATEX "2021groth-tops-children.lua")
%L forths["<-->"] = function () pusharrow("<-->") end
\pu

% «defs»  (to ".defs")

\long\def\ColorRed   #1{{\color{Red1}#1}}
\long\def\ColorViolet#1{{\color{MagentaVioletLight}#1}}
\long\def\ColorViolet#1{{\color{Violet!50!black}#1}}
\long\def\ColorGreen #1{{\color{SpringDarkHard}#1}}
\long\def\ColorGreen #1{{\color{SpringGreenDark}#1}}
\long\def\ColorGreen #1{{\color{SpringGreen4}#1}}
\long\def\ColorGray  #1{{\color{GrayLight}#1}}
\long\def\ColorGray  #1{{\color{black!30!white}#1}}
\long\def\ColorBrown #1{{\color{Brown}#1}}
\long\def\ColorBrown #1{{\color{brown}#1}}
\long\def\ColorOrange#1{{\color{orange}#1}}

\def\drafturl{http://angg.twu.net/LATEX/2020-2-C2.pdf}
\def\drafturl{http://angg.twu.net/2020.2-C2.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}

% (find-angg "LUA/defwithmanyargs.lua" "Bottle")

\def\und#1#2{\underbrace{#1}_{\text{#2}}}
\def\und#1#2{\underbrace{#1}_{#2}}

\def\sa#1#2{\expandafter\def\csname myarg#1\endcsname{#2}}
\def\ga#1{\csname myarg#1\endcsname}

\def\clop{\ovl{(·)}}

\def\Cst{\mathsf{CST}}
\def\Can{\mathsf{Can}}
\def\can{\mathsf{can}}
\def\Incs     {\mathsf{Incs}}
\def\inc      {\mathsf{inc}}
\def\CanSub   {\mathsf{CanSub}}
\def\SubPoints{\mathsf{SubPoints}}
\def\Subsets  {\mathsf{Subsets}}


% «defs-Bottle»  (to ".defs-Bottle")
% (find-LATEX "2021groth-tops-children.tex" "Bottle")
%
\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-7pt", scale="7pt", 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\Bo  #1{{       \BottleSetArgs{#1}\zha{Bottle-7pt}        }}
\def\bbo #1{{\left[ \BottleSetArgs{#1}\zha{Bottle-5pt} \right]}}
\def\pwbo#1{{\left( \BottleSetArgs{#1}\zha{Bottle-8pt} \right)}}



% «defs-GRO»  (to ".defs-GRO")
%L GRO_ts   = TCGSpec.new("45; 22, 24")
%L GRO_td_0 = TCGDims {h=15,  v=8,  q=15, crh=3.5,  crv=7, qrh=5}
%L GRO_td_2 = TCGDims {h=65, v=50,  q=15, crh=20,  crv=15, qrh=5}
%L GRO_tq   = TCGQ.newdsoa(GRO_td_0, GRO_ts,
%L                                  {tdef="GROSmall", meta="1pt s"},
%L                                  "h ap")
%L GRO_tq:LRputs("!ga{L1} !ga{L2} !ga{L3} !ga{L4}",
%L               "!ga{R1} !ga{R2} !ga{R3} !ga{R4} !ga{R5}"):output()
%L
%L GRO_tq   = TCGQ.newdsoa(GRO_td_2, GRO_ts,
%L                                  {tdef="GROBig", meta="1pt p"},
%L                                  "h v ap")
%L GRO_tq:LRputs("!ga{L1} !ga{L2} !ga{L3} !ga{L4}",
%L               "!ga{R1} !ga{R2} !ga{R3} !ga{R4} !ga{R5}"):output()
%
\pu
%
\def\GROSetargs#1#2#3#4#5#6#7#8#9{
             \sa{R5}{#9}%
  \sa{L4}{#4}\sa{R4}{#8}%
  \sa{L3}{#3}\sa{R3}{#7}%
  \sa{L2}{#2}\sa{R2}{#6}%
  \sa{L1}{#1}\sa{R1}{#5}}
%
\def\GRO#1#2#3#4#5#6#7#8#9{{%
  \GROSetargs{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}{#9}
  \tcg{GROSmall}}}
%
\def\GROBig#1#2#3#4#5#6#7#8#9{{%
  \GROSetargs{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}{#9}
  \tcg{GROBig}}}
%
\def\Gro#1{\left( \GRO#1 \right)}
\def\gro#1{       \GRO#1        }



\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\pmt         #1{\pmat{\text{#1}}}
\def\pmtt      #1#2{\pmat{\text{#1} \\ \text{#2}}}
\def\pmttt   #1#2#3{\pmat{\text{#1} \\ \text{#2} \\ \text{#3}}}
\def\pmtttt#1#2#3#4{\pmat{\text{#1} \\ \text{#2} \\ \text{#3} \\ \text{#4}}}

\def\smt         #1{\sm{\text{#1}}}
\def\smtt      #1#2{\sm{\text{#1} \\ \text{#2}}}
\def\smttt   #1#2#3{\sm{\text{#1} \\ \text{#2} \\ \text{#3}}}
\def\smtttt#1#2#3#4{\sm{\text{#1} \\ \text{#2} \\ \text{#3} \\ \text{#4}}}

\def\Sieveson{\mathsf{Sieves\_on}}
\def\Coveringsieveson{\mathsf{Covering\_sieves\_on}}
\def\Coveringsieveson{\mathsf{Covsieves\_on}}

\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\SetOXop{{\Set^{\OX^\op}}}
\def\SetCop {{\Set^{\catC^\op}}}

\def\rotl#1{\rotatebox{90}{$#1$}}
\def\rotr#1{\rotatebox{270}{$#1$}}

\def\DP{\calD(\bfP)}
\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${}''$}






%  _____ _ _   _
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/
%
% «title»  (to ".title")
% (grsp 1 "title")
% (grsa   "title")
% (c2m202p2p 1 "title")
% (c2m202p2    "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

\begin{tabular}{c}
  \bf \Large Grothendieck \\
  \bf \Large Topologies \\
  \bf \Large for Children \\
\end{tabular}

\bsk

Eduardo Ochs

% http://angg.twu.net/math-b.html#2021-groth-tops
{\footnotesize

\url{http://angg.twu.net/math-b.html\#2021-groth-tops}

}

\end{center}

\newpage


% «abstract»  (to ".abstract")
% (grsp 2 "abstract")
% (grsa   "abstract")

{\bf Abstract (1)}

\ssk

The notes in

\ssk

{% \footnotesize

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

}

\ssk

\noindent
--- I'll refer to them as ``\cite{FavC}'' from here on --- define an
extensible diagrammatic language that lets us take complex definitions
in Category Theory and then complement them with several kinds of
diagrams to lower the level of complexity and abstraction of the
original definition. What we usually get after adding these diagrams
is the original definition (very abstract, ``for adults'') drawn side
to side with diagrams for particular cases (``for children''), in two
parallel diagrams with the same shape; see the introduction of
\cite{FavC} for several different overviews of the method, and for
several different attempts to define ``children'' in a useful way.

\newpage

{\bf Abstract (2)}

\ssk

The definition of a Grothendieck topology is quite hard to understand
--- I found it {\sl impossible} for many years --- and in this
presentation I will show how the extend the diagrammatic language from
\cite{FavC} to handle that. Most of the material that I will present
is in

\ssk

\url{http://angg.twu.net/LATEX/2021groth-tops-children.pdf},

\ssk

\noindent
but I need to confess that this is an early draft that I need to
rewrite as soon as possible.

\ssk

The presentation will be in Portuguese, with slides in English.



\newpage

% «what-we-will-need»  (to ".what-we-will-need")

{\bf What we will need:}

1) The order topologies/ZHAs generated by 2-column graphs

from \cite[sections 14--17]{PH1},

%L H45_ts = TCGSpec.new("45; 22, 24")
%L H45_td = TCGDims {h=35,  v=28,  q=15, crh=7.5,  crv=9, qrh=5}
%L H45_tq = TCGQ.newdsoa(H45_td, H45_ts, {tdef="H45", meta="1pt p"}, "h v ap")
%L H45_tq:lrs():output()
%L H45_mp = H45_ts:mp({zdef="H45_zha", scale="11pt", meta=nil})
%L H45_mp:addlrs():output()
\pu

$$\scalebox{0.9}{$
  D = (P,A) = \tcg{H45}
  \qquad
  \Opens(D) = \Opens_A(P) = \zha{H45_zha}
  $}
$$



\newpage

{\bf What we will need (2)}

2) This extension to the notations in \cite{PH1}:

% (grcp 10 "down-sets")
% (grcp 12 "down-sets")
% (grc     "down-sets")
% (grc     "down-sets" "for sets of down-sets:")

$$\scalebox{0.8}{$
  \begin{array}{rcl} 
  \left[ \bo{0 000 1?? 111} \right]
    &=& \setofst{U ∈ \Opens(B)}{\text{$U$ is of the form $\bo{0 000 1?? 111}$}} \\
    &=& \setofst{U ∈ \Opens(B)}{\bo{0 000 100 111} ⊆ U ⊆ \bo{0 000 111 111}} \\
    &=& \left\{ \bo{0 000 100 111}, \;
                \bo{0 000 110 111}, \;
                \bo{0 000 111 111}  
        \right\} \\
  \\
  \left[ \bo{· ··· 1?? 111} \right]
    &=& \setofst{U ∈ \Opens(12)}{\text{$U$ is of the form $\bo{· ··· 1?? 111}$}} \\
    &=& \setofst{U ∈ \Opens(12)}{\bo{· ··· 100 111} ⊆ U ⊆ \bo{· ··· 111 111}} \\
    &=& \left\{ \bo{· ··· 100 111}, \;
                \bo{· ··· 110 111}, \;
                \bo{· ··· 111 111}  
        \right\} \\[18pt]
    &=& \{\; 
        {↓}\{10,02\}, \; 
        {↓}\{11,02\}, \;
        {↓}12
        \;\} \\ 
  \end{array}
  $}
$$


\newpage

{\bf What we will need (3)}

Several conventions from the beginning of \cite{FavC},

the convention on ``functors as objects'' from \cite[sec.7.12]{FavC},

and a new conventions for drawing diagrams of {\sl names},

{\sl pronounciations} and {\sl notations}.

\def\ru{\rotl{⊂}}

% (find-latexscan-links "C3" "20210517_hierarchy")
$
\myvcenter{
\includegraphics[height=4cm]{2020-2-C3/20210517_hierarchy.pdf}
}
\qquad
\setlength{\arraycolsep}{2.5pt}
\begin{matrix}
  U  &∈& \Opens(X) \\
  \ru && \ru \\
  V  &∈& \calS &∈& Ω(U) &=& \Downs({↓}U) \\
      &&        && \ru \\
   W &∈& \calU &∈& J(U) \\
\end{matrix}
$

\newpage

% «typical-letters»  (to ".typical-letters")
% «typical-values»  (to ".typical-values")
% (grsp 7 "typical-values")
% (grsa   "typical-values")

\def\GroTopDiagramOfTypicalLetters{{
  \setlength{\arraycolsep}{2.5pt}
  \begin{matrix}
    U  &∈& \Opens(X) \\
    \ru && \ru \\
    V  &∈& \calS &∈& Ω(U) &=& \Downs({↓}U) \\
        &&        && \ru \\
     W &∈& \calU &∈& J(U) \\
  \end{matrix}
  }}

\def\GroTopDiagramOfTypicalValues{{
  \setlength{\arraycolsep}{2.5pt}
  \def\OX {\left[ \bo{0 000 1?? 111} \right]}
  \def\OX {       \Bo{{32} {20}{21}{22} {10}{11}{12} {00}{01}{02}} }
  \def\OmU{\left[ \bo{· ··· ??? ???} \right]}
  \def\JU {\left[ \bo{· ··· 1?? 111} \right]}
  \def\cS {       \bo{· ··· 100 110}        }
  \def\cU {       \bo{· ··· 110 111}        }
  \def\dU {       \bo{· ··· 111 111}        }
  \begin{matrix}
    12  &∈& \OX \\
    \ru && \ru \\
    01  &∈& \cS &∈& \OmU &=& \Downs\left(\dU\right) &=& \Downs({↓}12) \\
        &&        && \ru \\
    10 &∈&  \cU &∈& \JU \\
  \end{matrix}
  }}


\vspace*{-0.8cm}

$$\begin{array}{c}
  \GroTopDiagramOfTypicalLetters
  \\
  \\
  \\
  \GroTopDiagramOfTypicalValues
  \end{array}
$$



% (favp 50 "functors-as-objects")
% (fav     "functors-as-objects")



\newpage

% «quotient-tops»  (to ".quotient-tops")
% (grsp 4 "quotient-tops")
% (grsa   "quotient-tops")

{\bf Quotient topologies}

Consider this partition of $\R$:
%
$$P \;\; = \;\;
  \{ \und{(-∞,1)}{A},
     \und{ [1,2)}{B},
     \und{ [2,3]}{C},
     \und{ (3,4]}{D},
     \und{(4,+∞)}{E}
  \}
$$

We will say that a subset $U⊆\R$ {\sl respects $P$} iff

for every $I∈P$ either $I⊂U$ or $I∩U=∅$.

For example, $B∪D = [1,2)∪(3,4]$ respects $P$,

but $[0.5,2.34]$ does not.

Let:

$$\begin{array}{rcl}
    \Pts_P(\R) &=& \setofst{U∈  \Pts(\R)}{\text{$U$ respects $P$}} \\
  \Opens_P(\R) &=& \setofst{U∈\Opens(\R)}{\text{$U$ respects $P$}} \\
  \end{array}
$$

\bsk

Then $\Pts_P(\R)$ has $2^5 = 32$ elements, and $\Opens_P ⊂ \Pts_P(\R)$.

\newpage

{\bf Quotient topologies (2)}

Here is another way to draw $P$ and the conditions

that an $U∈\Pts_P(\R)$ must obey to obey $U∈\Opens_P(\R)$:

% (c2m202somas2p 8 "imagens-de-intervalos")
% (c2m202somas2    "imagens-de-intervalos")

$$P \;\; = \;\;
  \{ \und{(-∞,1)}{A},
     \und{ [1,2)}{B},
     \und{ [2,3]}{C},
     \und{ (3,4]}{D},
     \und{(4,+∞)}{E}
  \}
$$
%
%D diagram ABCDE
%D 2Dx     100 +15 +15 +15 +15
%D 2D  100         C
%D 2D
%D 2D  +20     B       D
%D 2D
%D 2D  +20 A               E
%D 2D
%D ren A B C D E ==> \c{A} \c{B} \c{C} \c{D} \c{E}
%D
%D (( B A -> C B -> C D -> D E ->
%D
%D ))
%D enddiagram
%D
%$$\pu
%  \def\c#1{#1⊂U}
%  \diag{??}
%$$
%
$$\unitlength=20pt
 \celllower=2.5pt%
 %\def\cellfont{\scriptsize}%
 %
 \vcenter{\hbox{%
 \beginpicture(0,0)(5,4)
 \pictgrid%
 \pictpiecewise{(0,1)--(1,1)o
                (1,2)c--(2,2)o
                (2,3)c--(3,3)c
                (3,2)o--(4,2)c
                (4,1)o--(5,1)
               }%
 \put(0.5,1.4){\cell{A}}%
 \put(1.5,2.4){\cell{B}}%
 \put(2.5,3.4){\cell{C}}%
 \put(3.5,2.4){\cell{D}}%
 \put(4.5,1.4){\cell{E}}%
 \pictaxes%
 \end{picture}%
 }}
 %
 \qquad
 %
 \pu
 \def\c#1{#1⊂U}
 \diag{ABCDE}
$$

\newpage

{\bf Quotient topologies (3)}

Here are the 10 elements of $\Opens_P(\R)$:

$$\setlength{\arraycolsep}{0pt}
  \begin{matrix}
    & ABCDE & & & \\
    & & ABDE & & \\
    & ABD & & ADE & \\
 AB &   & AE &   & DE \\
    & A &    & E &    \\
    &   & ∅  &   &    \\
 \end{matrix}
$$


\newpage

% «2CGs»  (to ".2CGs")
% (grsp 7 "2CGs")
% (grsa   "2CGs")

{\bf 2-column graphs and their order topologies}

...or: 2CGs and ZHAs

% (grcp 4 "order-topologies")
% (grcp 9 "order-topologies")
% (grc    "order-topologies")
% (grcp 30 "ex-topologies")
% (grc     "ex-topologies")

%L H_ts = TCGSpec.new("32; 32,")
%L D_ts = TCGSpec.new("33; 32,")
%L H_td = TCGDims {h=35,  v=28,  q=15, crh=7.5,  crv=9, qrh=5}
%L D_td = TCGDims {h=35,  v=28,  q=15, crh=7.5,  crv=9, qrh=5}
%L H_tq = TCGQ.newdsoa(H_td, H_ts, {tdef="H", meta="1pt p"}, "h v ap")
%L D_tq = TCGQ.newdsoa(D_td, D_ts, {tdef="D", meta="1pt p"}, "h v ap")
%L H_tq:lrs():output()
%L D_tq:lrs():output()
%L H_mp = H_ts:mp({zdef="H_zha", scale="11pt", meta=nil})
%L D_mp = D_ts:mp({zdef="D_zha", scale="11pt", meta=nil})
%L H_mp:addlrs():output()
%L D_mp:addlrs():output()
\pu

$$X = H = \tcg{H}
  \qquad
  \Opens(X) = \Opens(H) = \zha{H_zha}
$$
%
$$D = N = \tcg{D}
  \qquad
  \Downs(D) = \Downs(N) = \zha{D_zha}
$$

\newpage

% «GRO»  (to ".GRO")
% (grsp 8 "GRO")
% (grsa   "GRO")

\vspace*{-1.25cm}

%L thistop   = v"46"
%L thisleft  = v"40"
%L thisright = v"06"
%L
%L Foo_ts   = TCGSpec.new("43; , ")
%L Foo_mp   = Foo_ts:mp({zdef="Foo_zha", scale="12pt", meta=nil})
%L Foo_mp.ap.s = "    "
%L Foo_mp:putcolors("00", "46", "~")
%L Foo_mp:putleftgen(2,  2)
%L Foo_mp:putrightgen(2, 5)
%L -- Foo_mp:putleftgen(2,  2, nil, "", "", "")
%L -- Foo_mp:putrightgen(2, 5, nil, "", "", "")
%L
%L Foo_mp:transfercolors()
%L Foo_mp:output()
\pu

\long\def\ColorGreen #1{{\color{SpringDarkHard}#1}}
\long\def\ColorGreen #1{{\color{SpringGreen4}#1}}
\long\def\ColorGreen #1{{\color{SpringGreenDark}#1}}

\def\GG  #1{\ColorGreen{#1}}
\def\G #1#2{\ColorGreen{#1#2}}
\def\RR  #1{\ColorRed  {#1}}
\def\R #1#2{\ColorRed  {#1#2}}
\def\OO  #1{\ColorOrange{#1}}
\def\O #1#2{\ColorOrange{#1#2}}
\def\YY  #1{{\color{yellow}#1}}
\def\Y #1#2{{\color{yellow}#1#2}}

$$\scalebox{1.75}{$
  \zha{Foo_zha}
  $}
$$


%R local A = 4/                      45                \
%R            |                  44      35            |
%R            |              43      34      25        |
%R            |          42      33    !G24    !O15    |
%R            |    !Y41      32      23    !R14    !Y05|
%R            |!Y40    !O31    !G22      13    !O04    |
%R            |    !Y30    !R21      12      03        |
%R            |        !O20      11      02            |
%R            |              10      01                |
%R            \                  00                    /
%R A:tomp({zdef="GRO", scale="12pt", meta=""}):addcells():output()
%R
%R local A = 4/                    !s45                \
%R            |                !s44    !s35            |
%R            |            !s43    !s34    !M25        |
%R            |        !s42    !s33    !M24    !M15    |
%R            |    !s41    !s32    !M23    !M14    !M05|
%R            |!s40    !M31    !M22    !s13    !M04    |
%R            |    !M30    !M21    !s12    !s03        |
%R            |        !M20    !s11    !s02            |
%R            |            !s10    !s01                |
%R            \                !s00                    /
%R A:tomp({zdef="GRO2", scale="30pt", meta=""}):addcells():output()

\pu


{
\sa{b04}{\OO{\gro{0000 11110}}}
\sa{b05}{\YY{\gro{0000 11111}}}
\sa{b14}{\RR{\gro{1000 11110}}}
\sa{b15}{\OO{\gro{1000 11111}}}
\sa{b20}{\OO{\gro{1100 00000}}}
\sa{b21}{\RR{\gro{1100 10000}}}
\sa{b22}{\GG{\gro{1100 11000}}}
\sa{b23}{   {\gro{1100 11100}}}
\sa{b24}{\GG{\gro{1100 11110}}}
\sa{b25}{   {\gro{1100 11111}}}
\sa{b30}{\YY{\gro{1110 00000}}}
\sa{b31}{\OO{\gro{1110 10000}}}
%
$\scalebox{1}{$
  \zha{GRO}
  $}
  %
  %a
  %
  \def\M#1#2{.#1#2.}
  \def\M#1#2{\ga{b#1#2}}
  \def\s#1#2{      }
  \scalebox{0.75}{$
  \hspace{-1cm}
  \zha{GRO2}
  $}
$
%
}



%$$\GROBig
%  {\bsh??111abcd}
%  {\bsh·?·1·abcd}  {\bsh··1·1}
%  {\bsh···1·}  {\bsh····?}
%$$

\newpage

% «intervals-top»  (to ".intervals-top")
% (grsp 13 "intervals-top")
% (grsa    "intervals-top")

% (find-LATEX "edrxgac2.tex" "beginpicture")
\def\aspictaxes#1{{\linethickness{0.5pt}#1}}
\def\aspictgrid#1{{\color{black!20!white}\linethickness{0.3pt}#1}}
\def\aspiecewise#1{{\linethickness{1pt}#1}}
\def\asintervals#1{
  \begin{picture}(5.4,4.4)(-0.2,-0.2)
    \aspictgrid{
      % Horizontal lines:
      \Line(-0.1,0)(5.1,0)
      \Line(-0.1,1)(5.1,1)
      \Line(-0.1,2)(5.1,2)
      \Line(-0.1,3)(5.1,3)
      \Line(-0.1,4)(5.1,4)
      % Vertical lines:
      \Line(0,4.1)(0,-0.1)
      \Line(1,4.1)(1,-0.1)
      \Line(2,4.1)(2,-0.1)
      \Line(3,4.1)(3,-0.1)
      \Line(4,4.1)(4,-0.1)
      \Line(5,4.1)(5,-0.1)
    }
    \aspictaxes{
      % Horizontal axis:
      \Line(-0.2,0)(5.2,0)
      % Horizontal axis, ticks:
      \Line(0,-0.2)(0,0.2)
      \Line(1,-0.2)(1,0.2)
      \Line(2,-0.2)(2,0.2)
      \Line(3,-0.2)(3,0.2)
      \Line(4,-0.2)(4,0.2)
      \Line(5,-0.2)(5,0.2)
      % Vertical axis:
      \Line(0,-0.2)(0,4.2)
      % Vertical axis, ticks:
      \Line(-0.2,0)(0.2,0)
      \Line(-0.2,1)(0.2,1)
      \Line(-0.2,2)(0.2,2)
      \Line(-0.2,3)(0.2,3)
      \Line(-0.2,4)(0.2,4)
    }
    \aspiecewise{#1}
  \end{picture}
  }


\def\Aseg{\polyline(-0.2,1)(1,1)}
\def\Bseg{\polyline(1,2)(2,2)}
\def\Cseg{\polyline(2,3)(3,3)}
\def\Dseg{\polyline(3,2)(4,2)}
\def\Eseg{\polyline(4,1)(5.2,1)}
\def\Adots{\put(1,1){\opendot}}
\def\Bdots{\put(1,2){\closeddot} \put(2,2){\opendot}}
\def\Cdots{\put(2,3){\closeddot} \put(3,3){\closeddot}}
\def\Ddots{\put(3,2){\opendot}   \put(4,2){\closeddot}}
\def\Edots{\put(4,1){\opendot}}

\sa{i00}{\asintervals{                             
                                                    }}
\sa{i01}{\asintervals{                        \Eseg
                                              \Edots}}
\sa{i02}{\asintervals{                  \Dseg \Eseg
                                        \Ddots\Edots}}
\sa{i10}{\asintervals{\Aseg                        
                      \Adots                        }}
\sa{i11}{\asintervals{\Aseg                   \Eseg
                      \Adots                  \Edots}}
\sa{i12}{\asintervals{\Aseg             \Dseg \Eseg
                      \Adots            \Ddots\Edots}}
\sa{i20}{\asintervals{\Aseg \Bseg                  
                      \Adots\Bdots                  }}
\sa{i21}{\asintervals{\Aseg \Bseg             \Eseg
                      \Adots\Bdots            \Edots}}
\sa{i22}{\asintervals{\Aseg \Bseg       \Dseg \Eseg
                      \Adots\Bdots      \Ddots\Edots}}
\sa{i32}{\asintervals{\Aseg \Bseg \Cseg \Dseg \Eseg
                      \Adots\Bdots\Cdots\Ddots\Edots}}


\unitlength=20pt
\unitlength=10pt

$$\def\I#1#2{\ga{i#1#2}}
  \begin{matrix}
     & \I32 \\
     && \I22 \\
     & \I21 && \I12 \\
     \I20 && \I11 && \I12 \\
     & \I10 && \I01 \\
     && \I00 \\
  \end{matrix}
$$

% (find-LATEX "2021groth-tops-children.tex" "Bottle")

\newpage

% «lindenhovius-filter»  (to ".lindenhovius-filter")
% (grsp 15 "lindenhovius-filter")
% (grsa    "lindenhovius-filter")
% (lindp 10 "2.4")
% (lindt 10 "2.4")
% (linda    "2.4")

\def\R#1{\ColorRed{#1}}



{\bf Lemma 2.4.} Let $J$ be a Grothendieck topology on $P$. Then $J(p)$ is a
filter of sieves on $p$ in the sense that:

\msk

a) $S∈J(p)$ implies $R∈J(p)$ for each sieve $R$ on $p$ containing $S$;

b) $S,R∈J(p)$ implies $S∩R∈J(p)$.

\msk

{\bf Proofs:}

\msk

a) Let $S∈J(p)$ and $R∈Ω(p)$ such that $S⊂R$. Then if $q∈S$, we
have $q∈R$, so $R∩{↓}q = {↓}q∈J(q)$. It follows now from the
transitivity axiom that $R∈J(p)$.

\msk

b) If $S,R∈J(p)$, and let $q∈R$. Then $(S∩{↓}R)∩{↓}q = S∩(R∩{↓}q) =
S∩{↓}q∈J(q)$ by the stability axiom. So $(S∩R)∩{↓}q∈J(q)$ for each
$q∈R$, hence by the transitivity axiom, it follows that $S∩R∈J(p)$.


\newpage

\def\H{\hspace*}

a) $S∈J(p)$ implies $R∈J(p)$ for each sieve $R$ on $p$ containing $S$;

{\bf Proof:}

a) Let $S∈J(p)$ and $R∈Ω(p)$ such that $S⊂R$. Then if $q∈S$, we have
$q∈R$, so $R∩{↓}q = {↓}q∈J(q)$. It follows now from the transitivity
axiom that $R∈J(p)$.

%:
%:            [q∈S]^1  S⊂R                   
%:            ------------                   
%:              \R{q∈R}      R∈Ω(p)           
%:              ---------   ------           
%:              {↓}q⊂{↓}R   {↓}R=R           
%:              ------------------   --------
%:                    {↓}q⊂R         (hasmax)
%:             ---------------      -------------
%:             \R{R∩{↓}q={↓}q}      \R{{↓}q∈J(q)}
%:             ------------------------------
%:                            R∩{↓}q∈J(q)
%:                          ----------------1  -------
%:  S∈J(p)  R∈Ω(p)\H{-2cm}  ∀q∈S.R∩{↓}q∈J(q)   (trans)
%:  --------------------------------------------------
%:           \R{R∈J(p)}
%:
%:           ^2.4a
%:
\pu
$$\scalebox{0.9}{$
  \ded{2.4a}
  $}
$$


\newpage

b) $S,R∈J(p)$ implies $S∩R∈J(p)$.

{\bf Proof:}

b) If $S,R∈J(p)$, and let $q∈R$. Then $(S∩{↓}R)∩{↓}q = S∩(R∩{↓}q) =
S∩{↓}q∈J(q)$ by the stability axiom. So $(S∩R)∩{↓}q∈J(q)$ for each
$q∈R$, hence by the transitivity axiom, it follows that $S∩R∈J(p)$.


%:
%:
%:                                      [q∈R]^1     R∈J(p)          
%:                                     ---------   -------          
%:                                     {↓}q⊂{↓}R   {↓}R=R        
%:                                     ---------------           
%:                                        {↓}q⊂R                 
%:                                       -----------               
%:                                       R∩{↓}q={↓}q               
%:                       ------          --------------------------------
%:   R∈J(p)   S∈J(p)     (stab)          \R{(S∩R)∩{↓}q=S∩(R∩{↓}q)=S∩{↓}q}
%:   ------   -----------------------    --------------------------------
%:   R⊂{↓}p   \R{∀q∈{↓}p.S∩{↓}q∈J(q)}       (S∩R)∩{↓}q=S∩{↓}q
%:   --------------------------------    ----------------------1
%:      ∀q∈R.S∩{↓}q∈J(q)                 ∀q∈R.(S∩R)∩{↓}q=S∩{↓}q
%:      -------------------------------------------------------   -------
%:                 \R{∀q∈R.(S∩R)∩{↓}q∈J(q)}                       (trans)
%:                 ------------------------------------------------------
%:                                  \R{S∩R∈J(p)}
%:
%:                                  ^2.4b
%:
\pu
$$\scalebox{0.8}{$
  \ded{2.4b}
  $}
$$

\newpage

%   ____ _               _  __       _                                     
%  / ___| | __ _ ___ ___(_)/ _|_   _(_)_ __   __ _   _ __ ___   __ _ _ __  
% | |   | |/ _` / __/ __| | |_| | | | | '_ \ / _` | | '_ ` _ \ / _` | '_ \ 
% | |___| | (_| \__ \__ \ |  _| |_| | | | | | (_| | | | | | | | (_| | |_) |
%  \____|_|\__,_|___/___/_|_|  \__, |_|_| |_|\__, | |_| |_| |_|\__,_| .__/ 
%                              |___/         |___/                  |_|    
%
% «classifying-map-1»  (to ".classifying-map-1")
% (grsp 18 "classifying-map-1")
% (grsa    "classifying-map-1")

\unitlength=1pt  % for the pullbacks

\vspace*{-1cm}


In a topos with inclusions $\catE = \Set^\catD$

the classifying map $χ_f$ of an inclusion $f: A \ito B$

is a map $χ_f:B→Ω$ such that for every $u∈\catD_0$

the map $χ_f(u):B(u)→Ω(u) = \Downs({↓}u)$ takes each $b∈B(u)$

to $\Cst(A{∩}B_{(u,b)}) ∈ Ω(u)$... where $B_{(u,b)}$ is defined by:

\msk

If $b∈B(u)$ then $B_{(u,b)}$ is the smallest canonical subobject

of $B$ having (or: ``generated by'') $B_{(u,b)}(u) = \{b\}$.

\msk

Diagram: 

\msk

%D diagram classifying-map-1
%D 2Dx     100  +45  +35  +35
%D 2D  100 B0 - A0 - A1 - A2
%D 2D      |    |    |     | 
%D 2D  +25 B1 - A3 - A4 - A5
%D 2D      |
%D 2D  +15 |    C0 - C1 - C2
%D 2D      |
%D 2D  +10 B2   D0 - D1 - D2
%D 2D
%D ren B0 B1 B2 ==> \Cst(A{∩}B_{(u,b)}) {↓}u=\Cst(B_{(u,b)}) 1
%D ren A0 A1 A2 ==> A{∩}B_{(u,b)} A 1
%D ren A3 A4 A5 ==> B_{(u,b)} B Ω
%D ren C0 C1 C2 ==> B_{(u,b)}(u) B(u) Ω(u)
%D ren D0 D1 D2 ==> b b χ_f(u)(b)
%D
%D (( A0 A1 `-> A1 A2  -> .plabel= a !
%D    A0 A3 `-> A1 A4 `-> .plabel= l f A2 A5 `-> .plabel= r ⊤
%D    A3 A4 `-> A4 A5  -> .plabel= b χ_f 
%D    A0 A3 `->
%D    A0 relplace 9 8 \pbsymbol{7}
%D    A1 relplace 7 7 \pbsymbol{7}
%D    B2 xy+= 0 0
%D    B0 A0 <-> # B0 A3 >->
%D    B0 B1 `->
%D    B1 A3 <-> B1 B2 `-> A3 B2 >->
%D
%D    C0 C1 `-> C1 C2 `->
%D    D0 D1 |-> D1 D2 |->
%D ))
%D enddiagram
%D
$\pu
  \diag{classifying-map-1}
$


\newpage

%  _     _ _   _   _      _   _ 
% | |   (_) |_| |_| | ___| \ | |
% | |   | | __| __| |/ _ \  \| |
% | |___| | |_| |_| |  __/ |\  |
% |_____|_|\__|\__|_|\___|_| \_|
%                               
% «LittleN»  (to ".LittleN")
% (grsp 18 "LittleN")
% (grsa    "LittleN")

\makeatletter
\def\LittleNSetArgs#1{\LittleNArgs@#1}
\def\LittleNArgs@#1#2#3#4{%
  \sa{L2}{#1}\sa{R2}{#2}%
  \sa{L1}{#3}\sa{R1}{#4}%
  }
\makeatother

%L LittleN_ts = TCGSpec.new("22; 21,")
%L LittleN_td = TCGDims {h=30,  v=28,  q=15, crh=10,  crv=9, qrh=5}
%L LittleN_tq = TCGQ.newdsoa(LittleN_td, LittleN_ts, {tdef="LittleN 2CG", meta="1pt p"}, "h v ap")
%L LittleN_tq:lrs():output()
%L LittleN_mp = LittleN_ts:mp({zdef="LittleN ZHA", scale="11pt", meta=nil})
%L LittleN_mp:addlrs():output()
\pu

%L LittleN_ts   = TCGSpec.new("22; 21,")
%L LittleN_td_0 = TCGDims {h=12,  v=8,  q=15, crh=3,   crv=7,  qrh=3}
%L LittleN_td_1 = TCGDims {h=25, v=17,  q=15, crh=8,   crv=5,  qrh=10}
%L LittleN_td_2 = TCGDims {h=45, v=45,  q=15, crh=15,  crv=16, qrh=5}
%L LittleN_tq   = TCGQ.newdsoa(LittleN_td_0, LittleN_ts,
%L                                  {tdef="LittleNSmall", meta="1pt s"},
%L                                  "h ap")
%L LittleN_tq:LRputs("!ga{L1} !ga{L2}",
%L                   "!ga{R1} !ga{R2}"):output()
%L
%L LittleN_tq   = TCGQ.newdsoa(LittleN_td_1, LittleN_ts,
%L                                  {tdef="LittleNMedium", meta="1pt s p"},
%L                                  "h v ap")
%L LittleN_tq:LRputs("!ga{L1} !ga{L2}",
%L                   "!ga{R1} !ga{R2}"):output()
%L
%L LittleN_tq   = TCGQ.newdsoa(LittleN_td_2, LittleN_ts,
%L                                  {tdef="LittleNBig", meta="1pt p"},
%L                                  "h v ap")
%L LittleN_tq:LRputs("!ga{L1} !ga{L2}",
%L                   "!ga{R1} !ga{R2}"):output()

\pu
\def\littlena  #1{{       \LittleNSetArgs{#1}\tcg{LittleNSmall}        }}
\def\littlen   #1{{       \LittleNSetArgs{#1}\tcg{LittleNMedium}       }}
\def\littlenbig#1{{       \LittleNSetArgs{#1}\tcg{LittleNBig}          }}

% «OLittleN»  (to ".OLittleN")

\makeatletter
\def\OLittleNSetArgs#1{\OLittleNArgs@#1}
\def\OLittleNArgs@#1#2#3#4#5#6#7#8{%
             \sa{21}{#1}\sa{22}{#2}%
  \sa{10}{#3}\sa{11}{#4}\sa{12}{#5}%
  \sa{00}{#6}\sa{01}{#7}\sa{02}{#8}%
  }
\makeatother

%R local OLittleN = 7/       !ga{22}              \
%R                   |!ga{21}       !ga{12}       |
%R                   |       !ga{11}       !ga{02}|
%R                   |!ga{10}       !ga{01}       |
%R                   \       !ga{00}              /
%R OLittleN:tomp({zdef="OLittleNSmall", scale="6pt", meta="t"}):addcells():output()

\pu
\def\olittlen  #1{{       \OLittleNSetArgs{#1}\zha{OLittleNSmall}        }}


% «classifier-LittleN»  (to ".classifier-LittleN")

\def\Aa{\littlena{0011}}
\def\Ab{\littlena{1011}}
\def\Ac{\littlena{1111}}
%
\def\Ba{\littlen{     ∅         ∅     {\{5\}} {\{6\}} }}
\def\Bb{\littlen{   {\{2\}}     ∅     {\{5\}} {\{6\}} }}
%
\def\Ca{\littlen{     ∅       {\{4\}} {\{5\}} {\{6\}} }}
\def\Cb{\littlen{ {\{1,2\}} {\{3,4\}} {\{5\}} {\{6\}} }}
%
\def\Da{\littlen{ {\{21\}} {\{20\}} {\{10\}} {\{01\}} }}
\def\Db{\littlen{ {↓21} {↓20} {↓10} {↓01} }}
\def\Db{\littlenbig{
  {\olittlen{{21}{·} {10}{11}{·} {00}{01}{·}}}
  {\olittlen{ {·}{·}  {·}{·}{·} {00}{01}{02}}}
  {\olittlen{ {·}{·} {10}{·}{·}   {00}{·}{·}}}
  {\olittlen{ {·}{·}  {·}{·}{·}  {00}{01}{·}}}
  }}

\def\chif{
  \begin{array}{cc}
  \chi_f(2▁)=\csm{(1,10),\\(2,10)}, &
  \chi_f(▁2)=\csm{(3,01),\\(4,20)}, \\[5pt]
  \chi_f(1▁)=\csm{(5,10)}, &
  \chi_f(▁1)=\csm{(6,01)} \\
  \end{array}
}


% ----------------------------------------

Let's look at a particular case, with:

$$\catD = \scalebox{0.6}{$\tcg{LittleN 2CG}$},
  \quad
  \text{so:}
  \quad
  H = \Sub({\littlena{1111}}) = \;\; \scalebox{0.6}{$\zha{LittleN ZHA}$}
$$


%D diagram PB-LittleN
%D 2Dx     100  +60
%D 2D  100 A1 - A2
%D 2D      |     | 
%D 2D  +45 A4 - A5
%D 2D
%D ren A1 A2 ==> \Ca \Da
%D ren A4 A5 ==> \Cb \Db
%D
%D (( A1 A2  -> .plabel= a !
%D    A1 A4 `-> .plabel= l f A2 A5 `-> .plabel= r ⊤
%D    A4 A5  -> .plabel= b χ_f 
%D    A1 relplace 15 12 \pbsymbol{7}
%D ))
%D enddiagram
%D
\pu

% \vspace*{-1.2cm}

$$\scalebox{0.8}{$
    \diag{PB-LittleN}
    $}
  \quad
  \def\mini#1{\scalebox{0.7}{$#1$}}
  \begin{array}{rcl}
    χ_f(2▁)(2) &=& \Cst(A∩B_{(2▁,2)}) \\
               &=& \Cst( \mini{\Ca} ∩ \mini{\Bb} ) \\
               &=& \Cst( \mini{\Ba} ) \\
               &=& \littlena{0011} \\
               &=& 11.
  \end{array}
$$




\newpage








%D diagram classifier-LittleN
%D 2Dx     100  +40  +50  +60
%D 2D  100 B0 - A0 - A1 - A2
%D 2D      |    |    |     | 
%D 2D  +45 B1 - A3 - A4 - A5
%D 2D      |
%D 2D  +35 B2
%D 2D
%D ren B0 A0 A1 A2 ==> \Aa \Ba \Ca \Da
%D ren B1 A3 A4 A5 ==> \Ab \Bb \Cb \Db
%D ren B2          ==> \Ac
%D
%D (( A0 A1 `-> A1 A2  -> .plabel= a !
%D    A0 A3 `-> A1 A4 `-> .plabel= l f A2 A5 `-> .plabel= r ⊤
%D    A3 A4 `-> A4 A5  -> .plabel= b χ_f 
%D    A0 A3 `->
%D    A0 relplace 15 12 \pbsymbol{7}
%D    A1 relplace 15 12 \pbsymbol{7}
%D    B2 xy+= 0 0
%D    B0 A0 <-> # B0 A3 >->
%D    B0 B1 `->
%D    B1 A3 <-> B1 B2 `-> A3 B2 >->
%D
%D    newnode: chif at: @A4+v(20,40)
%D      .TeX= \scalebox{0.8}{$\chif$} place
%D ))
%D enddiagram
%D
\vspace*{-1.2cm}

$$\pu
  %\scalebox{0.9}{$
    \diag{classifier-LittleN}
  %$}
$$


\newpage

% «bijections-1»  (to ".bijections-1")
% (grsp 21 "bijections-1")
% (grsa    "bijections-1")

%D diagram ??
%D 2Dx     100       +50       +70
%D 2D  100 \aqmarks  \asubset \anucleus
%D 2D
%D 2D  +40                    \aclop
%D 2D
%D 2D  +45           \agrtop \alttop
%D 2D
%D # ren ==>
%D
%D (( \aqmarks \asubset  <-->
%D    \asubset \anucleus <-> .plabel= b \LindC
%D    \agrtop  \alttop   <-> .plabel= b \MM
%D    \alttop  \aclop    <-> .plabel= r \CLT
%D    \asubset \agrtop   <-> .plabel= l \LindC
%D    \anucleus \aclop   <-->
%D    \aqmarks \anucleus <-> .slide= 20pt .plabel= a \PHdois
%D ))
%D enddiagram
%D
$$\pu
  \def\asubset {\pmtt{a subset}{$\calY⊆\catD_0$}}
  \def\anucleus{\pmtt{a nucleus}{$(·)^*:H→H$}}
  \def\agrtop  {\pmttt{a Grothendieck}{topology}{$J⊂Ω$}}
  \def\alttop  {\pmtttt{a Lawvere-}{Tierney}{topology}{$j:Ω→Ω$}}
  \def\aclop   {\pmttt{a closure operator:}{for every $E∈\catE$ a}{$\clop_E:\Incs(E)→\Incs(E)$}}
  \def\aqmarks {\pmtttt{a set of}{question}{marks}{$Q⊆\catD_0$}}
  % (lindp 74 "C.4")
  % (linda    "C.4")
  % (find-books "__cats/__cats.el" "maclane-moerdijk")
  % (find-books "__cats/__cats.el" "mclarty")
  \def\MM      {\smtttt{\cite{MacLaneMoerdijk}}{Sec V.4}{Thm 1}{p.233}}
  \def\LindC   {\smttt{\cite{Lindenhovius}}{Thm C4,}{p.74}}
  \def\CLT     {\smtt{\cite[sec.21]{McLarty},}{\cite[sec.2.6]{ClopsAndTops}}}
  \def\PHdois  {\smt{\cite{PH2}}}
  \diag{??}
$$


%\cite{Lindenhovius}
%\cite{MacLaneMoerdijk}
%\cite{ClopsAndTops}
%\cite{McLarty}


\newpage

\printbibliography

\GenericWarning{Success:}{Success!!!}  % Used by `M-x cv'

\end{document}

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make -f 2019.mk STEM=2021groth-tops-children-slides veryclean
make -f 2019.mk STEM=2021groth-tops-children-slides pdf

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