|
Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (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}
% __ __ _
% | \/ | __ _| | _____
% | |\/| |/ _` | |/ / _ \
% | | | | (_| | < __/
% |_| |_|\__,_|_|\_\___|
%
% <make>
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2021groth-tops-children-slides veryclean
make -f 2019.mk STEM=2021groth-tops-children-slides pdf
% Local Variables:
% coding: utf-8-unix
% ee-tla: "grs"
% End: