% (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:
