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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (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: