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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-LATEX "2026jacobs.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2026jacobs.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2026jacobs.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2026jacobs.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2026jacobs.pdf"))
% (defun e () (interactive) (find-LATEX "2026jacobs.tex"))
% (defun o () (interactive) (find-LATEX "2026jacobs.tex"))
% (defun u () (interactive) (find-latex-upload-links "2026jacobs"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2026jacobs.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (defun oe () (interactive) (find-2a '(o) '(e)))
% (code-eec-LATEX "2026jacobs")
% (find-pdf-page "~/LATEX/2026jacobs.pdf")
% (find-sh0 "cp -v ~/LATEX/2026jacobs.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2026jacobs.pdf /tmp/pen/")
% (find-xournalpp "/tmp/2026jacobs.pdf")
% file:///home/edrx/LATEX/2026jacobs.pdf
% file:///tmp/2026jacobs.pdf
% file:///tmp/pen/2026jacobs.pdf
% http://anggtwu.net/LATEX/2026jacobs.pdf
% https://anggtwu.net/LATEX/2026jacobs.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-cps "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-anggs "Caepro5 Piecewise2 Maxima2")
% (find-MM-aula-links "2026jacobs" "2" "jac2026" "jac")
% «.geometry» (to "geometry")
% «.edrx26a» (to "edrx26a")
% «.biber» (to "biber")
% «.edrx26b» (to "edrx26b")
% «.edrx26c» (to "edrx26c")
% «.defs» (to "defs")
% «.footer» (to "footer")
% «.defs-T-and-B» (to "defs-T-and-B")
%
% «.title» (to "title")
% «.toc» (to "toc")
% «.links» (to "links")
% «.cartesianness-jacobs» (to "cartesianness-jacobs")
% «.cartesianness-edrx» (to "cartesianness-edrx")
% «.fibred-equality» (to "fibred-equality")
% «.BCC-equality» (to "BCC-equality")
%
% «.writetoc» (to "writetoc")
% «.references» (to "references")
% «.make-with-bib» (to "make-with-bib")
% ;-- 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-LATEX "dednat7-test1.tex")
%\usepackage{proof} % For derivation trees ("%:" lines)
\input diagxy % For 2D diagrams ("%D" lines)
%\xyoption{curve} % For the ".curve=" feature in 2D diagrams
%
% «geometry» (to ".geometry")
% (find-es "tex" "geometry")
\usepackage[a6paper, landscape,
top=1.5cm, bottom=.25cm, left=1cm, right=1cm, includefoot
]{geometry}
%
% «edrx26a» (to ".edrx26a")
\usepackage{edrx26a} % (find-LATEX "edrx26a.sty")
%
% «biber» (to ".biber")
\usepackage[backend=biber,
style=alphabetic]{biblatex} % (find-es "tex" "biber")
\addbibresource{catsem-ab.bib} % (find-LATEX "catsem-ab.bib")
\addbibresource{education.bib} % (find-LATEX "education.bib")
%
\begin{document}
% «edrx26b» (to ".edrx26b")
\input edrx26b.tex % (find-LATEX "edrx26b.tex")
% «edrx26c» (to ".edrx26c")
% (find-LATEX "edrx26c.tex")
%L processsubfile "edrx26c.tex" -- runs the "%L"s
\input edrx26c % loads the defs
% «defs» (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
% (find-LATEX "edrx21.sty")
% «footer» (to ".footer")
% (find-LATEX "edrxheadfoot.tex")
\def\drafturl{http://anggtwu.net/LATEX/2026-1-C2.pdf}
\def\drafturl{http://anggtwu.net/2026.1-C2.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}
% «defs-T-and-B» (to ".defs-T-and-B")
\long\def\ColorDarkOrange#1{{\color{orange!90!black}#1}}
\def\T(Total: #1 pts){{\bf(Total: #1)}}
\def\T(Total: #1 pts){{\bf(Total: #1 pts)}}
\def\T(Total: #1 pts){\ColorRed{\bf(Total: #1 pts)}}
\def\B (#1 pts){\ColorDarkOrange{\bf(#1 pts)}}
\def\Eq {\mathsf{Eq}}
\def\BCCL {\mathsf{BCCL}}
% ;-- title
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (jac2026p 1 "title")
% (jac2026a "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Notes about Jacobs's book}
\bsk
%Aula nn: ponha o título aqui
%
%\bsk
Eduardo Ochs - RCN/PURO/UFF
Psicopata do CEFET
\url{https://anggtwu.net/math-b.html}
\end{center}
%\newpage
% ;-- toc
% «toc» (to ".toc")
% (to "writetoc")
% ;-- links
% «links» (to ".links")
% (jac2026p 2 "links")
% (jac2026a "links")
%{\bf Links}
%
%\scalebox{0.6}{\def\colwidth{16cm}\firstcol{
%}\anothercol{
%}}
% (jac2020p)
% (jac2020a)
% (find-books "__cats/__cats.el" "freyd76")
% (find-books "__cats/__cats.el" "freyd-scedrov" "28" "1.39. The language of diagrams")
% (find-books "__cats/__cats.el" "heller-tierney" "55" "Freyd")
% (misp 17 "freyd-notation")
% (misa "freyd-notation")
% (misa "freyd-notation" "equalizers")
% (misp 18 "freyd-with-functors")
% (misa "freyd-with-functors")
% (misp 44 "ness")
% (misa "ness")
\newpage %-- cartesianness-jacobs
%
% ____ _ _ _
% / ___|__ _ _ __| |_ | | __ _ ___ ___ | |__ ___
% | | / _` | '__| __| _ | |/ _` |/ __/ _ \| '_ \/ __|
% | |__| (_| | | | |_ | |_| | (_| | (_| (_) | |_) \__ \
% \____\__,_|_| \__| \___/ \__,_|\___\___/|_.__/|___/
%
% «cartesianness-jacobs» (to ".cartesianness-jacobs")
% (jac2026p 99 "cartesianness-jacobs")
% (jac2026a "cartesianness-jacobs")
% (find-books "__cats/__cats.el" "jacobs" "27" "Finally, we come to the definition of 'fibration'")
\SLIDE{Cartesianness (Jacobs)}
\scalebox{0.8}{\def\colwidth{14cm}\firstcol{
From \cite[p.27]{Jacobs}:
\ssk
{\bf 1.1.3. Definition.} Let $p:\bbE→\bbB$ be a functor.
(i) A morphism $f:X→Y$ in $\bbE$ is {\bf Cartesian over} $u:I→J$ in
$\bbB$ if $pf=u$ and every $g:Z→Y$ in $\bbE$ for which one has
$pg=u∘w$ for some $w:pZ→I$, uniquely determines an $h:Z→X$ in $\bbE$
above $w$ with $f∘h=g$. In a diagram:
%
%L forths["-"] = function () pusharrow("-") end
%
%D diagram cartesianness-jacobs
%D 2Dx 100 +20 +20 +30
%D 2D 100 A0 ____
%D 2D +15 E \ \
%D 2D +10 | A1 - A2
%D 2D |
%D 2D +10 v A3 ____
%D 2D +15 B \ \
%D 2D +10 A4 - A5
%D
%D ren E B A0 A1 A2 A3 A4 A5 ==> \bbE \bbB Z X Y pZ I J
%D
%D (( E B -> .plabel= l p
%D
%D A0 A1 --> .plabel= l h
%D A1 A2 -> .plabel= b f
%D A0 A2 -> .plabel= a g
%D A3 A4 -> .plabel= l w
%D A4 A5 -> .plabel= b u
%D A3 A5 -> .plabel= a u∘w=pg
%D ))
%D enddiagram
\pu
$$\diag{cartesianness-jacobs}
$$
We call $f:X→Y$ in the total category $\bbE$ Cartesian if it is
Cartesian over its underlying map $pf$ in $\bbB$.
}\anothercol{
}}
\newpage %-- cartesianness-edrx
% ____ _ _____ _
% / ___|__ _ _ __| |_ | ____|__| |_ ____ __
% | | / _` | '__| __| | _| / _` | '__\ \/ /
% | |__| (_| | | | |_ | |__| (_| | | > <
% \____\__,_|_| \__| |_____\__,_|_| /_/\_\
%
% «cartesianness-edrx» (to ".cartesianness-edrx")
% (jac2026p 99 "cartesianness-edrx")
% (jac2026a "cartesianness-edrx")
\SLIDE{Cartesianness (Edrx)}
%L forths["-"] = function () pusharrow("-") end
%
%D diagram cartesianness-edrx
%D 2Dx 100 +00 +20 +15 +15 +10 +20 +15 +15 +10 +20
%D 2D 100 U0 U1
%D 2D | |
%D 2D +10 A0 ____ | B0 ____ | C0 ____
%D 2D \ \ | \ \ | \ \
%D 2D +15 A1 - A2 | B1 - B2 | C1 - C2
%D 2D | |
%D 2D +10 A3 ____ | B3 ____ | C3 ____
%D 2D \ \ | \ \ | \ \
%D 2D +15 A4 - A5 | B4 - B5 | C4 - C5
%D 2D | |
%D 2D +10 L0 L1
%D
%D ren A0 A1 A2 A3 A4 A5 ==> {} Q R {} {} {}
%D ren U0 L0 B0 B1 B2 B3 B4 B5 ==> ∀ {} P Q R A B C
%D ren U1 L1 C0 C1 C2 C3 C4 C5 ==> ∃! {} P Q R A B C
%D
%D (( A1 A2 -> .plabel= a β
%D
%D U0 L0 -
%D
%D # B0 B1 -> .plabel= l α
%D B1 B2 -> .plabel= b β
%D B0 B2 -> .plabel= a γ
%D B3 B4 -> .plabel= l f
%D B4 B5 -> .plabel= b g
%D B3 B5 -> .plabel= a h
%D
%D U1 L1 -
%D
%D C0 C1 -> .plabel= l α
%D C1 C2 -> .plabel= b β
%D C0 C2 -> .plabel= a γ
%D C3 C4 -> .plabel= l f
%D C4 C5 -> .plabel= b g
%D C3 C5 -> .plabel= a h
%D ))
%D enddiagram
\pu
\scalebox{0.8}{\def\colwidth{9cm}\firstcol{
Let $p:\bbE \to \bbB$ be our projection functor.
$β:Q→R$ is {\sl cartesian} when:
%
$$\diag{cartesianness-edrx}
$$
The details:
%
$$∀\pmat{P,A,B,C,
γ,f,g,h,\\
A=pP,B=pQ,C=pR,\\
g=pβ,h=pγ,h=g∘f
}.
∃!\pmat{α,\\
γ=β∘α,\\
f=pα}
$$
}\anothercol{
}}
\newpage %-- fibred-equality
% «fibred-equality» (to ".fibred-equality")
% (jac2026p 99 "fibred-equality")
% (jac2026a "fibred-equality")
% (find-books "__cats/__cats.el" "jacobs" "190" "3.4. Fibred equality")
\SLIDE{Fibred equality}
\sa {<id,pi'>} {〈\id,π'〉}
\sa {IJ} {I×J}
\sa {KJ} {K×J}
\sa {IJJ} {(I×J)×J}
\sa {KJJ} {(K×J)×J}
\sa {EIJ} {\bbE_{I×J}}
\sa {EIJJ} {\bbE_{(I×J)×J}}
\sa {dd(I,J)} {{δ(I,J)}}
\sa {dd(K,J)} {{δ(K,J)}}
\sa {dd(I,J)*} {{δ(I,J)^*}}
\sa {dd(K,J)*} {{δ(K,J)^*}}
\sa {Eq(I,J)} {{\Eq(I,J)}}
\sa {Eq(K,J)} {{\Eq(K,J)}}
\sa {Eq_IJ} {{\Eq_{I,J}}}
\sa {Eq_KJ} {{\Eq_{K,J}}}
\sa {Ud(I,J)} {{\coprod_\ga{dd(I,J)}}}
\sa {UE(I,J)} {\mat{\ga {Eq_IJ} P =\\
\ga {Ud(I,J)} P }}
%D diagram p190-Eq
%D 2Dx 100 +65 +40 +60
%D 2D 100 A0 |-> A1 B0 |-> B1
%D 2D | | | |
%D 2D v v v v
%D 2D +30 A2 <-| A3 B2 <-| B3
%D 2D
%D 2D +20 EIJ <=> EIJJ
%D 2D +20 IJ --> IJJ IJ' --> IJJ'
%D 2D
%D ren A0 A1 ==> P \ga{UE(I,J)}
%D ren A2 A3 ==> \ga{dd(I,J)*}Q Q
%D ren IJ IJJ ==> \ga{IJ} \ga{IJJ}
%D ren EIJ EIJJ ==> \ga{EIJ} \ga{EIJJ}
%D
%D ren B0 B1 ==> P(i,j) j{=}j'∧P(i,j)
%D ren B2 B3 ==> Q(i,j,j) Q(i,j,j')
%D ren IJ' IJJ' ==> \ga{IJ} \ga{IJJ}
%D
%D (( A0 A1 |->
%D A2 A3 <-|
%D A0 A3 harrownodes nil 20 nil <->
%D A0 A2 ->
%D A1 A3 ->
%D
%D EIJ EIJJ -> sl^ .plabel= a \ga{Eq_IJ}=\ga{Ud(I,J)}
%D EIJ EIJJ <- sl_ .plabel= b \ga{dd(I,J)*}
%D IJ IJJ -> .plabel= a \ga{dd(I,J)}=\ga{<id,pi'>}
%D
%D B0 B1 |->
%D B2 B3 <-|
%D B0 B3 harrownodes nil 20 nil <->
%D B0 B2 ->
%D B1 B3 ->
%D
%D IJ' IJJ' -> .plabel= a (i,j)↦((i,j),j)
%D ))
%D enddiagram
\pu
\scalebox{0.9}{\def\colwidth{9cm}\firstcol{
\vspace*{0.1cm}
{\bf 3.4. Fibred equality}
(From {\cite[p.190]{Jacobs}}):
%
$$\diag{p190-Eq}
$$
}\anothercol{
}}
\newpage %-- BCC-equality
% «BCC-equality» (to ".BCC-equality")
% (jac2026p 99 "BCC-equality")
% (jac2026a "BCC-equality")
\SLIDE{Beck-Chevally for equality}
%D diagram TP
%D 2Dx 100
%D 2D 100 B4
%D 2D ^
%D 2D |
%D 2D v
%D 2D +20 B6
%D 2D
%D 2D EKJJ
%D 2D
%D 2D +15 KJJ
%D 2D
%D ren B4 B6 ==> \ga{Eq(K,J)}{f'}^*P f^*\ga{Eq(I,J)}P
%D # ren EKJJ ==> \bbE_{(K×J)×J}
%D ren KJJ ==> (K×J)×J
%D
%D (( B4 B6 -> sl_ .plabel= l ♮
%D B4 B6 <- sl^ .plabel= r \BCCL
%D # EKJJ place
%D KJJ place
%D ))
%D enddiagram
\pu
\sa{width1}{9cm}
\sa{width2}{7cm}
\scalebox{0.6}{\def\colwidth{\ga{width1}}\firstcol{
{}
From \cite[p.191]{Jacobs}:
...the Beck-Chevalley condition holds: for each map $u:K→I$ in $\bbB$
(between the parameter objects) the canonical natural transformation
%
$$\ga{Eq(K,J)} (u×\id)^* \Longrightarrow ((u×\id)×\id)^* \ga{Eq(I,J)}$$
%
is an isomorphism.
\bsk
In the diagrams of the following pages we have $f' = u×\id$ and
$f = (u×\id)×\id$, so the canonical natural transformation above is:
%
$$T : \ga{Eq(K,J)} {f'}^* \Longrightarrow f^* \ga{Eq(I,J)}$$
If we apply it to an object $P$ we get the morphism
%
$$TP : \ga{Eq(K,J)} {f'}^* P \to f^* \ga{Eq(I,J)} P$$
in the fiber $\bbE_{(K×J)×J}$, that is drawn as `$♮$' in my diagrams
-- as in the column at the right.
}\def\colwidth{\ga{width2}}\anothercol{
{}
$$\scalebox{2.0}{$
\diag{TP}
$}
$$
}}
\sa {B0} {B0}
\sa {B1} {B1}
\sa {B2} {B2}
\sa {B2'} {B2'}
\sa {B3} {B3}
\sa {B4} {B4}
\sa {B5} {B5}
\sa {B6} {B6}
\sa {B7} {B7}
\sa {b0} {b0}
\sa {b1} {b1}
\sa {b2} {b2}
\sa {b3} {b3}
\sa {f} {f}
\sa {f'} {f'}
\sa {z} {z}
\sa {z'} {z'}
%D diagram BCCL-generic
%D 2Dx 100 +70 +70 +70
%D 2D 100 B0 <====================== B1
%D 2D -\\ -\\
%D 2D | \\ | \\
%D 2D v \\ v \\
%D 2D +20 B2 <\\> B2' ============== B3 \\
%D 2D /\ \/ /\ \/
%D 2D +15 \\ B4 \\ B5
%D 2D \\ - \\ -
%D 2D \\ | \\|
%D 2D \\v \v
%D 2D +20 B6 <===================== B7
%D 2D
%D 2D +10 b0 |---------------------> b1
%D 2D |-> |->
%D 2D +35 b2 |--------------------> b3
%D 2D
%D ren B0 B1 ==> \ga{B0} \ga{B1}
%D ren B2 B2' B3 ==> \ga{B2} \ga{B2'} \ga{B3}
%D ren B4 B5 ==> \ga{B4} \ga{B5}
%D ren B6 B7 ==> \ga{B6} \ga{B7}
%D ren b0 b1 ==> \ga{b0} \ga{b1}
%D ren b2 b3 ==> \ga{b2} \ga{b3}
%D ((
%D B0 B1 <-| B0 B2 -> B0 B2' -> B1 B3 -> B2 B2' <-> B2' B3 <-|
%D B0 B4 |-> B1 B5 |->
%D B2 B6 <-| B3 B7 <-|
%D B6 B7 <-| B5 B7 -> .plabel= r \id
%D B4 B6 -> sl_ .plabel= l ♮ B4 B6 <- sl^ .plabel= r \BCCL
%D B0 B2' midpoint B1 B3 midpoint harrownodes nil 20 nil <-|
%D B0 B2 midpoint B4 B6 midpoint dharrownodes nil 20 nil |->
%D B1 B3 midpoint B5 B7 midpoint dharrownodes nil 20 nil <-|
%D ))
%D ((
%D b0 b1 -> .plabel= b \ga{f'}
%D b0 b2 -> .plabel= l \ga{z'}
%D b1 b3 -> .plabel= r \ga{z}
%D b2 b3 -> .plabel= a \ga{f}
%D b0 relplace 20 7 \pbsymbol{7}
%D ))
%D enddiagram
\pu
\sa {BCCL-equality-jacobs} {{
%
\sa {B1} {P}
\sa {B5} {\ga{Eq_IJ} P}
\sa {B7} {\ga{Eq_IJ} P}
\sa {B3} {z^* \ga{Eq_IJ} P}
\sa {B2'} {{f'}^* z^* \ga{Eq_IJ} P}
%
\sa {B6} {f^* \ga{Eq_IJ} P}
\sa {B2} {{z'}^* f^* \ga{Eq_IJ} P}
%
\sa {B0} {{f'}^* P}
\sa {B4} {\ga{Eq_KJ} {f'}^* P}
%
\sa {b0} {\ga{KJ}} \sa {b1} {\ga{IJ}}
\sa {b2} {\ga{KJJ}} \sa {b3} {\ga{IJJ}}
%
\sa {f'} {f' = u×\id}
\sa {z} {z = \ga{dd(I,J)}}
\sa {z'} {z' = \ga{dd(K,J)}}
\sa {f} {f = (u×\id)×\id}
%
\diag{BCCL-generic}
}}
\sa {BCCL-equality-edrx} {{
%
\sa {B1} {P(i,j)}
\sa {B5} {j{=}j' ∧ P(i,j)}
\sa {B7} {j{=}j' ∧ P(i,j)}
\sa {B3} {j{=}j ∧ P(i,j)}
\sa {B2'} {j{=}j ∧ P(u(k),j)}
%
\sa {B6} {j{=}j' ∧ P(u(k),j)}
\sa {B2} {j{=}j ∧ P(u(k),j)}
%
\sa {B0} {P(u(k),j)}
\sa {B4} {j{=}j' ∧ P(u(k),j)}
%
\sa {b0} {\ga{KJ}} \sa {b1} {\ga{IJ}}
\sa {b2} {\ga{KJJ}} \sa {b3} {\ga{IJJ}}
%
\sa {f'} { (k,j) ↦ (u(k),j)}
\sa {z} {\;\; (i,j) ↦ ((i,j),j)}
\sa {z'} { (k,j) ↦ ((k,j),j) \;\;}
\sa {f} {((k,j),j') ↦ ((u(k),j),j')}
%
\diag{BCCL-generic}
}}
\scalebox{0.58}{\def\colwidth{18cm}\firstcol{
\vspace*{-2cm}
$$\ga{BCCL-equality-jacobs}
$$
\bsk
$$\ga{BCCL-equality-edrx}
$$
\vspace*{-2cm}
}\anothercol{
}}
%D diagram BCCL-std
%D 2Dx 100 +45 +55 +45
%D 2D 100 B0 <====================== B1
%D 2D -\\ -\\
%D 2D | \\ | \\
%D 2D v \\ v \\
%D 2D +20 B2 <\\> B2' ============== B3 \\
%D 2D /\ \/ /\ \/
%D 2D +15 \\ B4 \\ B5
%D 2D \\ - \\ -
%D 2D \\ | \\|
%D 2D \\v \v
%D 2D +20 B6 <===================== B7
%D 2D
%D 2D +10 b0 |---------------------> b1
%D 2D |-> |->
%D 2D +35 b2 |--------------------> b3
%D 2D
%D ((
%D B0 .tex= f^{\prime*}P B1 .tex= P
%D B2 .tex= z^{\prime*}f^*Σ_zP B2' .tex= f^{\prime*}z^*Σ_zP B3 .tex= z^*Σ_zP
%D B4 .tex= Σ_{z'}f^{\prime*}P B5 .tex= Σ_zP
%D B6 .tex= f^*Σ_zP B7 .tex= Σ_zP
%D B0 B1 <-| B0 B2 -> B0 B2' -> B1 B3 -> B2 B2' <-> B2' B3 <-|
%D B0 B4 |-> B1 B5 |->
%D B2 B6 <-| B3 B7 <-|
%D B6 B7 <-| B5 B7 -> .plabel= r \id
%D B4 B6 -> sl_ .plabel= l ♮ B4 B6 <- sl^ .plabel= r \BCCL
%D B0 B2' midpoint B1 B3 midpoint harrownodes nil 20 nil <-|
%D B0 B2 midpoint B4 B6 midpoint dharrownodes nil 20 nil |->
%D B1 B3 midpoint B5 B7 midpoint dharrownodes nil 20 nil <-|
%D ))
%D (( b0 .tex= X×_{Y}Z b1 .tex= Z b2 .tex= X b3 .tex= Y
%D b0 b1 -> .plabel= b f'
%D b0 b2 -> .plabel= l z'
%D b1 b3 -> .plabel= r z
%D b2 b3 -> .plabel= a f
%D b0 relplace 20 7 \pbsymbol{7}
%D ))
%D enddiagram
\pu
$$\diag{BCCL-std}$$
% ;-- writetoc
% «writetoc» (to ".writetoc")
\directlua{toclines:writetoc()}
% Writes in: (find-LATEXfile "2026jacobs.mytoc")
% See: (to "toc")
% ;-- references
% «references» (to ".references")
\printbibliography
% ;-- write-dnt-file
% «write-dnt-file» (to ".write-dnt-file")
% (find-fline "~/LATEX/" "2026jacobs.dnt")
% (find-fline "~/LATEX/2026jacobs.dnt")
%L write_dnt_file ("2026jacobs.dnt")
\pu
\GenericWarning{Success:}{Success!!!} % Used by `M-x cv'
\end{document}
% (find-pdfpages2-links "~/LATEX/" "2026jacobs")
% ;-- make-with-bib
% «make-with-bib» (to ".make-with-bib")
% (wiya "make-with-bib")
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
cd ~/LATEX/
# which biber
# biber --version
make -f 2019.mk STEM=2026jacobs veryclean
lualatex 2026jacobs.tex
biber 2026jacobs
lualatex 2026jacobs.tex
# (find-pdf-page "~/LATEX/2026jacobs.pdf")
% Local Variables:
% coding: utf-8-unix
% outline-regexp: "% +;--"
% ee-tla: "jac"
% ee-tla: "jac2026"
% End: