% (find-LATEX "2020adjunctions.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2020adjunctions.tex" :end))
% (defun d () (interactive) (find-pdf-page "~/LATEX/2020adjunctions.pdf"))
% (defun e () (interactive) (find-LATEX "2020adjunctions.tex"))
% (defun u () (interactive) (find-latex-upload-links "2020adjunctions"))
% (find-pdf-page   "~/LATEX/2020adjunctions.pdf")
% (find-sh0 "cp -v  ~/LATEX/2020adjunctions.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2020adjunctions.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2020adjunctions.pdf
%               file:///tmp/2020adjunctions.pdf
%           file:///tmp/pen/2020adjunctions.pdf
% http://angg.twu.net/LATEX/2020adjunctions.pdf
% (find-LATEX "2019.mk")

\documentclass[oneside]{book}
\usepackage[colorlinks,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")
%
% (find-es "tex" "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")
% \pu


%D diagram adjunction-labels
%D 2Dx     100     +30  
%D 2D  100 LA' <-| A'    
%D 2D      |       |    
%D 2D      v       v    
%D 2D  +30 LA <--| A    
%D 2D      |       |       
%D 2D      v       v       
%D 2D  +30 B |---> RB   
%D 2D      |       |    
%D 2D      v       v    
%D 2D  +30 B' |--> RB'   
%D 2D                  
%D 2D  +20 \catB  \catA    
%D
%D (( LA' A' <-|
%D    LA  A  <-|
%D    B  RB  |->
%D    B' RB' |->
%D
%D    LA' A harrownodes nil 20 nil <-| sl^
%D    LA RB harrownodes nil 20 nil <-| sl^ .plabel= a ♭
%D    LA RB harrownodes nil 20 nil |-> sl_ .plabel= b ♯
%D    B RB' harrownodes nil 20 nil |-> sl^
%D
%D    LA' LA -> .plabel= l Lf
%D     A'  A -> .plabel= r  f
%D    LA   B -> .plabel= l \sm{h^♭\\g}
%D     A  RB -> .plabel= r \sm{h\\g^♯}
%D     B  B' -> .plabel= l k
%D    RB RB' -> .plabel= r Rk
%D
%D    \catB \catA <- sl^ .plabel= a L
%D    \catB \catA -> sl_ .plabel= b R
%D ))
%D enddiagram
%
$$\pu
  \diag{adjunction-labels}
  \qquad
  \begin{array}{rcl}
  h^{♭♯} &=& h \\
  g^{♯♭} &=& g \\
  f;g^♯;Rk &=& (Lf;g;k)^♯ \\
  Lf;h^♭;k &=& (f;h;Rk)^♭ \\
  \end{array}
$$


% <defmateight>
% Skel: (find-defcsprefix-links "mateight" "MATRIX 4x2")
%
\def\defmateight#1#2{\expandafter\def\csname mateight-#1\endcsname{#2}}
\def\ifmateightundefined#1{\expandafter\ifx\csname mateight-#1\endcsname\relax}
\def\mateight#1{\ifmateightundefined{#1}
    \errmessage{UNDEFINED MATRIX 4x2: #1}
  \else
    \csname mateight-#1\endcsname
  \fi
}

\def\ph{\phantom}
\def\ms{\mathstrut}

\defmateight{A,B}  {\psm{\ms   &   \\
                         \ms   & A \\
                         \ms B &   \\
                         \ms   &   \\}}

\defmateight{A',B'}{\psm{\ms    & A' \\
                         \ms    &    \\
                         \ms    &    \\
                         \ms B' &    \\}}

\defmateight{LA->B}{\psm{\ms    &        \\
                         \ms LA & \ph{A} \\
                         \ms B  &        \\
                         \ms    &        \\}}

\defmateight{LA'->B'}{\psm{\ms LA' &        \\
                           \ms     & \ph{A} \\
                           \ms     &        \\
                           \ms B'  &        \\}}

\defmateight{A->RB}{\psm{\ms        &    \\
                         \ms        & A  \\
                         \ms \ph{B} & RB \\
                         \ms        &    \\}}

\defmateight{A'->RB'}{\psm{\ms        & A' \\
                           \ms        &    \\
                           \ms \ph{B} &    \\
                           \ms        & RB' \\}}



%D diagram sqcond-adj-1
%D 2Dx     100      +40            +50         +40     +45
%D 2D  100 AB    \Hom(LA,B) --> \Hom(A,RB)     g |---> gsh
%D 2D       |        |               |         -        -
%D 2D       |        |               |         |        v
%D 2D  +42  v        v               v         v      (gsh)'
%D 2D  +8  A'B'  \Hom(LA',B') -> \Hom(A',RB')  g' |-> (g')sh
%D 2D
%D 2D  +30        \Hom(L-,-) -> \Hom(-,R-)
%D 2D
%D ren           AB A'B'         ==>     \mateight{A,B} \mateight{A',B'}
%D ren   \Hom(LA,B) \Hom(A,RB)   ==>   \mateight{LA->B} \mateight{A->RB}
%D ren \Hom(LA',B') \Hom(A',RB') ==> \mateight{LA'->B'} \mateight{A'->RB'}
%D ren gsh (gsh)' ==>    g^♯ f;g^♯;Rk
%D ren g' (g')sh  ==> Lf;g;k (Lf;g;k)^♯
%D
%D (( AB A'B' -> .plabel= l (f,k)
%D
%D      \Hom(LA,B) \Hom(A,RB)   -> .plabel= a \sharp_{A,B}
%D      \Hom(LA,B) \Hom(LA',B') -> .plabel= l Lf;-;k   # \Hom(Lf,k)
%D      \Hom(A,RB) \Hom(A',RB') -> .plabel= l f;-;Rk   # \Hom(f,Rk)
%D    \Hom(LA',B') \Hom(A',RB') -> .plabel= a \sharp_{A',B'}
%D
%D    \Hom(L-,-) \Hom(-,R-) -> .plabel= a \sharp
%D
%D    g gsh |-> gsh (gsh)' |->
%D    g g' |-> g' (g')sh |->
%D ))
%D enddiagram
%D
$$\pu
  \diag{sqcond-adj-1}
$$

%D diagram sqcond-adj-2
%D 2Dx     100      +40            +50          +40       +45
%D 2D  100 AB    \Hom(LA,B) <-- \Hom(A,RB)      hfl <---| h 
%D 2D       |        |               |           -        -
%D 2D       |        |               |           v        |
%D 2D  +42  v        v               v         (hfl)'     v
%D 2D  +8  A'B'  \Hom(LA',B') <- \Hom(A',RB')  (h')fl <-| h'
%D 2D
%D 2D  +30        \Hom(L-,-) <- \Hom(-,R-)
%D 2D
%D ren           AB A'B'         ==>     \mateight{A,B} \mateight{A',B'}
%D ren   \Hom(LA,B) \Hom(A,RB)   ==>   \mateight{LA->B} \mateight{A->RB}
%D ren \Hom(LA',B') \Hom(A',RB') ==> \mateight{LA'->B'} \mateight{A'->RB'}
%D ren hfl (hfl)' ==>    h^♭ Lf;h^♭;k
%D ren h' (h')fl  ==> f;h;Rk (f;h;Rk)^♭
%D
%D (( AB A'B' -> .plabel= l (f,k)
%D
%D      \Hom(LA,B) \Hom(A,RB)   <- .plabel= a \flat_{A,B}
%D      \Hom(LA,B) \Hom(LA',B') -> .plabel= l Lf;-;k   # \Hom(Lf,k)
%D      \Hom(A,RB) \Hom(A',RB') -> .plabel= l f;-;Rk   # \Hom(f,Rk)
%D    \Hom(LA',B') \Hom(A',RB') <- .plabel= a \flat_{A',B'}
%D
%D    \Hom(L-,-) \Hom(-,R-) <- .plabel= a \flat
%D
%D    h hfl |-> hfl (hfl)' |->
%D    h h' |-> h' (h')fl |->
%D ))
%D enddiagram
%D
$$\pu
  \diag{sqcond-adj-2}
$$


\newpage

Putting an $\id$ at one extremity:
%:
%:  -------------------           -------------------    
%:  f;g^♯;Rk=(Lf;g;k)^♯           Lf;h^♭;k=(f;h;Rk)^♭    
%:  -----------------------       -------------------    
%:  f;g^♯;R\id=(Lf;g;\id)^♯       L\id;h^♭;k=(\id;h;Rk)^♭
%:  -----------------------       -------------------    
%:  f;g^♯=(Lf;g)^♯                h^♭;k=(h;Rk)^♭
%:                                                       
%:  ^r1                           ^r2                          
%:                                                       
%:  -------------------           -------------------    
%:  f;g^♯;Rk=(Lf;g;k)^♯           Lf;h^♭;k=(f;h;Rk)^♭    
%:  -----------------------       -----------------------
%:  \id;g^♯;Rk=(L\id;g;k)^♯       Lf;h^♭;\id=(f;h;R\id)^♭
%:  -----------------------       -----------------------
%:  g^♯;Rk=(g;k)^♯                Lf;h^♭=(f;h)^♭         
%:
%:  ^r3                           ^r4
%:
\pu
$$\ded{r1} \qquad \ded{r2}$$
$$\ded{r3} \qquad \ded{r4}$$
  

Putting an $\id$ at the middle:
%:
%:  -------------------           -------------------    
%:  f;g^♯;Rk=(Lf;g;k)^♯           Lf;h^♭;k=(f;h;Rk)^♭    
%:  -----------------------       -------------------    
%:  f;\id^♯;Rk=(Lf;\id;k)^♯       Lf;\id^♭;k=(f;\id;Rk)^♭    
%:  -----------------------       -------------------    
%:  f;\id^♯;Rk=(Lf;k)^♯           Lf;\id^♭;k=(f;Rk)^♭    
%:                                                       
%:  ^mid1                         ^mid2                          
%:                                                       
\pu
$$\ded{mid1} \qquad \ded{mid2}$$
  
Putting an $\id$ at one extremity:
%:
%:  -------------------           -------------------    
%:  f;g^♯;Rk=(Lf;g;k)^♯           Lf;h^♭;k=(f;h;Rk)^♭    
%:  -----------------------       -------------------    
%:  f;\id^♯;R\id=(Lf;\id;\id)^♯   L\id;\id^♭;k=(\id;\id;Rk)^♭
%:  -----------------------       -------------------    
%:  f;\id^♯=(Lf)^♯                \id^♭;k=(Rk)^♭
%:                                                       
%:  ^m1                           ^m2                          
%:                                                       
%:  -------------------           -------------------    
%:  f;g^♯;Rk=(Lf;g;k)^♯           Lf;h^♭;k=(f;h;Rk)^♭    
%:  -----------------------       -----------------------
%:  \id;\id^♯;Rk=(L\id;\id;k)^♯   Lf;h^♭;\id=(f;h;R\id)^♭
%:  -----------------------       -----------------------
%:  \id^♯;Rk=k^♯                  Lf;h^♭=(f;h)^♭         
%:
%:  ^r3                           ^r4
%:
  
  
  
  

  
  
  
  
  
  




\end{document}

% Local Variables:
% coding: utf-8-unix
% ee-tla: "adj"
% End:
