|
Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-LATEX "2026cwm.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2026cwm.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2026cwm.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page "~/LATEX/2026cwm.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2026cwm.pdf"))
% (defun e () (interactive) (find-LATEX "2026cwm.tex"))
% (defun o () (interactive) (find-LATEX "2026cwm.tex"))
% (defun u () (interactive) (find-latex-upload-links "2026cwm"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2026cwm.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (defun oe () (interactive) (find-2a '(o) '(e)))
% (code-eec-LATEX "2026cwm")
% (find-pdf-page "~/LATEX/2026cwm.pdf")
% (find-sh0 "cp -v ~/LATEX/2026cwm.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2026cwm.pdf /tmp/pen/")
% (find-xournalpp "/tmp/2026cwm.pdf")
% file:///home/edrx/LATEX/2026cwm.pdf
% file:///tmp/2026cwm.pdf
% file:///tmp/pen/2026cwm.pdf
% http://anggtwu.net/LATEX/2026cwm.pdf
% https://anggtwu.net/LATEX/2026cwm.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-cps "Caepro5 Piecewise2 Maxima2 DiagForth1")
% (find-Deps1-anggs "Caepro5 Piecewise2 Maxima2")
% (find-MM-aula-links "2026cwm" "2" "cwm2026" "cwm")
% «.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")
% «.defs-adj-names» (to "defs-adj-names")
% «.defs-adj-diag» (to "defs-adj-diag")
% «.defs-adj-minipage» (to "defs-adj-minipage")
% «.defs-abcells» (to "defs-abcells")
% «.defs-adjs-conj-pure» (to "defs-adjs-conj-pure")
% «.defs-adjs-conj-mixed» (to "defs-adjs-conj-mixed")
% «.defs-adjs-conj-sigma-tau» (to "defs-adjs-conj-sigma-tau")
% «.defs-conj-cell-eqs» (to "defs-conj-cell-eqs")
% «.adjunctions» (to "adjunctions")
% «.adjunctions-conj-pure» (to "adjunctions-conj-pure")
% «.adjunctions-conj-mixed» (to "adjunctions-conj-mixed")
%
% «.writetoc» (to "writetoc")
% «.references» (to "references")
% ;-- 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)}}
%L require "DiagForth1" -- (find-angg "LUA/DiagForth1.lua")
\pu
% ;-- title
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (cwm2026p 1 "title")
% (cwm2026a "title")
\thispagestyle{empty}
\begin{center}
\vspace*{1.2cm}
{\bf \Large Notes about Mac Lane's CWM}
\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")
% (cwm2026p 2 "links")
% (cwm2026a "links")
%{\bf Links}
%
%\scalebox{0.6}{\def\colwidth{16cm}\firstcol{
%}\anothercol{
%}}
% «defs-adj-names» (to ".defs-adj-names")
\sa {adjunction: Edrx names} {
\sa {A} {A}
\sa {B} {B}
\sa {L} {L}
\sa {R} {R}
\sa {LA} {LA}
\sa {RB} {RB}
\sa {catA} {\catA}
\sa {catB} {\catB}
\sa {sharp} {♯}
\sa {flat} {♭}
}
\sa {adjunction: CWM names} {
\sa {A} {x}
\sa {B} {a}
\sa {L} {F}
\sa {R} {G}
\sa {LA} {Fx}
\sa {RB} {Ga}
\sa {catA} {X}
\sa {catB} {A}
\sa {sharp} {φ}
\sa {flat} {ψ}
}
% «defs-adj-diag» (to ".defs-adj-diag")
%D diagram adjunction-1
%D 2Dx 100 +25
%D 2D 100 LA <======= A
%D 2D | |
%D 2D | <--> |
%D 2D v v
%D 2D +25 B =======> RB
%D 2D
%D 2D +10 catB <=> catA
%D 2D
%D ren LA A ==> \ga{LA} \ga{A}
%D ren B RB ==> \ga{B} \ga{RB}
%D ren catB catA ==> \ga{catB} \ga{catA}
%D
%D (( LA A <-|
%D LA B ->
%D A RB ->
%D B RB |->
%D
%D LA RB harrownodes nil 20 nil <- sl^ .plabel= a \ga{flat}
%D LA RB harrownodes nil 20 nil -> sl_ .plabel= b \ga{sharp}
%D
%D catB catA <- sl^ .plabel= a \ga{L}
%D catB catA -> sl_ .plabel= b \ga{R}
%D ))
%D enddiagram
\pu
\sa {adjunction diag Edrx} {{\ga{adjunction: Edrx names} \diag{adjunction-1}}}
\sa {adjunction diag CWM} {{\ga{adjunction: CWM names} \diag{adjunction-1}}}
% «defs-adj-minipage» (to ".defs-adj-minipage")
%
\Sa {adjunction minipage} {{
\begin{minipage}[t]{\colwidth}
\par Let $\ga{catB}$ and $\ga{catA}$ be categories.
\par An adjunction from $\ga{catB}$ to $\ga{catA}$
\par is a triple $〈\ga{L},\ga{R},\ga{sharp}〉:\ga{catA} \rightharpoonup \ga{catB}$
\par where $\ga{L}$ and $\ga{R}$ are functors
%
$$\ga{catA} \two/->`<-/^{\ga{L}}_{\ga{R}} \ga{catB}
$$
\par while $\ga{sharp}$ is a function
%
$$\ga{sharp}=\ga{sharp}_{\ga{A},\ga{B}}:
\ga{catB}(\ga{LA},\ga{B}) ≅
\ga{catA}(\ga{A},\ga{RB})
$$
\par which is natural in $\ga{A}$ and $\ga{B}$.
\par Some authors write $\ga{L}⊣\ga{R}$.
\end{minipage}
}}
%
\sa {adjunction minipage Edrx} {{
\def\colwidth{8cm}
\ga{adjunction: Edrx names}
\ga{adjunction minipage}
}}
\sa {adjunction minipage CWM} {{
\def\colwidth{8cm}
\ga{adjunction: CWM names}
\ga{adjunction minipage}
}}
\def\Dn#1{{\scriptstyle \Downarrow \, #1}}
% «defs-abcells» (to ".defs-abcells")
% (cwm2026a "defs-abcells")
% (find-angg "LUA/DiagForth1.lua" "relplace")
%L
%L forths["ab!"] = function ()
%L node_a = ds:pick(1)
%L node_b = ds:pick(0)
%L end
%L forths["ab@"] = function ()
%L ds:push(node_a)
%L ds:push(node_b)
%L end
%L
%L forths[".curve^^"] = function () forths[".curve="]("^32pt") end
%L forths[".curve^"] = function () forths[".curve="]( "^8pt") end
%L forths[".curve_"] = function () forths[".curve="]( "_8pt") end
%L forths[".curve__"] = function () forths[".curve="]("_32pt") end
%L
%L forthe["ab-name:"] = "e,e,w"
%L forths["ab-name:"] = function (dx,dy,TeX)
%L forths["ab@"]()
%L forths["midpoint"]()
%L forths["relplace"](dx,dy,TeX)
%L end
%L
%L y_eta = -10
%L y_eps = 10
%L
%L forthe["ab-name-up:"] = "w"
%L forthe["ab-name-mid:"] = "w"
%L forthe["ab-name-down:"] = "w"
%L forths["ab-name-up:"] = function (tex) forths["ab-name:"](0,y_eta,tex) end
%L forths["ab-name-mid:"] = function (tex) forths["ab-name:"](0,0, tex) end
%L forths["ab-name-down:"] = function (tex) forths["ab-name:"](0,y_eps,tex) end
%L
%L forths["ab-up-eta"] = function () dxyrun "ab-name-up: \\Dn{η}" end
%L forths["ab-up-eta'"] = function () dxyrun "ab-name-up: \\Dn{η'}" end
%L forths["ab-down-eps"] = function () dxyrun "ab-name-down: \\Dn{ε}" end
%L forths["ab-down-eps'"] = function () dxyrun "ab-name-down: \\Dn{ε'}" end
%L forths["ab-mid-id"] = function () dxyrun "ab-name-mid: \\Dn{\\id}" end
%L forths["ab-mid-sigma"] = function () dxyrun "ab-name-mid: \\Dn{σ}" end
%L forths["ab-mid-tau"] = function () dxyrun "ab-name-mid: \\Dn{τ}" end
%L
%L forths["ab-mid-id-L"] = function () dxyrun "ab-name-mid: \\Dn{\\id_L}" end
%L forths["ab-mid-id-R"] = function () dxyrun "ab-name-mid: \\Dn{\\id_R}" end
%L forths["ab-mid-id-L'"] = function () dxyrun "ab-name-mid: \\Dn{\\id_{L'}}" end
%L forths["ab-mid-id-R'"] = function () dxyrun "ab-name-mid: \\Dn{\\id_{R'}}" end
%L
%L forths["ab-curve^^"] = function () dxyrun "ab@ -> .curve^^" end
%L forths["ab-curve__"] = function () dxyrun "ab@ -> .curve__" end
%L
%L forths["ab-eta"] = function () dxyrun "ab-curve^^ ab-up-eta" end
%L forths["ab-eta'"] = function () dxyrun "ab-curve^^ ab-up-eta'" end
%L forths["ab-eps"] = function () dxyrun "ab-curve__ ab-down-eps" end
%L forths["ab-eps'"] = function () dxyrun "ab-curve__ ab-down-eps'" end
%L
%L forths["ab-sigma"] = function ()
%L dxyrun "ab@ -> .curve^ .plabel= a L"
%L dxyrun "ab@ => .curve_ .plabel= b L'"
%L dxyrun "ab-name-mid: \\Dn{σ}"
%L end
%L forths["ab-tau"] = function ()
%L dxyrun "ab@ => .curve^ .plabel= a R'"
%L dxyrun "ab@ -> .curve_ .plabel= b R"
%L dxyrun "ab-name-mid: \\Dn{τ}"
%L end
% «defs-adjs-conj-pure» (to ".defs-adjs-conj-pure")
% The two adjunctions - the "pure" diagrams:
%
% ___________ ___________
% / \ / \
% / ⇓ η \ / \
% / \ L / \ L
% A -----> B --R--> A -----> B = A -----> B ⇓id A -----> B
% L \ / L \ /
% \ ⇓ ε / \ /
% \___________/ \___________/
%
% ___________ ___________
% / \ / \
% / ⇓ η \ / \
% R / \ R / \
% B -----> A --L--> B -----> A = B -----> A ⇓id B -----> A
% \ / R \ / R
% \ ⇓ ε / \ /
% \___________/ \___________/
%
%
%
% ___________ ___________
% / \ / \
% / ⇓ η' \ / \
% / \ L' / \ L'
% A =====> B ==R'=> A =====> B = A =====> B ⇓id A =====> B
% L' \ / L' \ /
% \ ⇓ ε' / \ /
% \___________/ \___________/
%
% ___________ ___________
% / \ / \
% / ⇓ η \ / \
% R' / \ R' / \
% B =====> A ==L'=> B ======> A = B =====> A ⇓id B ======> A
% \ / R' \ / R'
% \ ⇓ ε / \ /
% \___________/ \___________/
%D diagram cells-pure-nw-se-1
%D 2Dx 100 +25 +25 +25
%D 2D 100 ___________
%D 2D / \
%D 2D / ⇓ η \
%D 2D / \ L
%D 2D 100 A0 ----> B1 -R--> A2 ----> B3
%D 2D L \ /
%D 2D \ ⇓ ε /
%D 2D \___________/
%D 2D
%D ren A0 B1 A2 B3 ==> \catA \catB \catA \catB
%D
%D (( A0 B1 -> .plabel= b L
%D B1 A2 -> .plabel= m R
%D A2 B3 -> .plabel= a L
%D A0 A2 ab! ab-eta
%D B1 B3 ab! ab-eps
%D ))
%D enddiagram
\pu
%D diagram cells-pure'-nw-se-1
%D 2Dx 100 +25 +25 +25
%D 2D 100 ___________
%D 2D / \
%D 2D / ⇓ η' \
%D 2D / \ L'
%D 2D 100 A0 ====> B1 =R'=> A2 ====> B3
%D 2D L' \ /
%D 2D \ ⇓ ε' /
%D 2D \___________/
%D 2D
%D ren A0 B1 A2 B3 ==> \catA \catB \catA \catB
%D
%D (( A0 B1 => .plabel= b L
%D B1 A2 => .plabel= m R
%D A2 B3 => .plabel= a L
%D A0 A2 ab! ab-eta'
%D B1 B3 ab! ab-eps'
%D ))
%D enddiagram
\pu
%D diagram cells-pure-nw-se-2
%D 2Dx 100 +25 +25 +25
%D 2D 100 ___________
%D 2D / \
%D 2D / \
%D 2D / \ L
%D 2D 100 A0 ----> B1 ⇓ id A2 ----> B3
%D 2D L \ /
%D 2D \ /
%D 2D \___________/
%D 2D
%D ren A0 B1 A2 B3 ==> \catA \catB \catA \catB
%D
%D (( A0 B1 -> .plabel= b L
%D A2 B3 -> .plabel= a L
%D A0 A2 ab! ab-curve^^
%D B1 A2 ab! ab-mid-id-L
%D B1 B3 ab! ab-curve__
%D ))
%D enddiagram
\pu
%D diagram cells-pure'-nw-se-2
%D 2Dx 100 +25 +25 +25
%D 2D 100 ___________
%D 2D / \
%D 2D / \
%D 2D / \ L'
%D 2D 100 A0 ====> B1 ⇓ id A2 ====> B3
%D 2D L' \ /
%D 2D \ /
%D 2D \___________/
%D 2D
%D ren A0 B1 A2 B3 ==> \catA \catB \catA \catB
%D
%D (( A0 B1 => .plabel= b L'
%D A2 B3 => .plabel= a L'
%D A0 A2 ab! ab-curve^^
%D B1 A2 ab! ab-mid-id-L'
%D B1 B3 ab! ab-curve__
%D ))
%D enddiagram
\pu
%D diagram cells-pure-ne-sw-1
%D 2Dx 100 +25 +25 +25
%D 2D 100 __________
%D 2D / \
%D 2D / ⇓ η \
%D 2D R / \
%D 2D 100 B0 ----> A1 -L--> B2 ----> A3
%D 2D \ / R
%D 2D \ ⇓ ε /
%D 2D \___________/
%D 2D
%D ren B0 A1 B2 A3 ==> \catB \catA \catB \catA
%D
%D (( B0 A1 -> .plabel= a R
%D A1 B2 -> .plabel= m L
%D B2 A3 -> .plabel= b R
%D B0 B2 ab! ab-eps
%D A1 A3 ab! ab-eta
%D ))
%D enddiagram
\pu
%D diagram cells-pure'-ne-sw-1
%D 2Dx 100 +25 +25 +25
%D 2D 100 __________
%D 2D / \
%D 2D / ⇓ η' \
%D 2D R' / \
%D 2D 100 B0 ====> A1 =L==> B2 ====> A3
%D 2D \ / R
%D 2D \ ⇓ ε' /
%D 2D \___________/
%D 2D
%D ren B0 A1 B2 A3 ==> \catB \catA \catB \catA
%D
%D (( B0 A1 => .plabel= a R'
%D A1 B2 => .plabel= m L'
%D B2 A3 => .plabel= b R'
%D B0 B2 ab! ab-eps'
%D A1 A3 ab! ab-eta'
%D ))
%D enddiagram
\pu
%D diagram cells-pure-ne-sw-2
%D 2Dx 100 +25 +25 +25
%D 2D 100 __________
%D 2D / \
%D 2D / \
%D 2D R / \
%D 2D 100 B0 ----> A1 ⇓id B2 ----> A3
%D 2D \ / R
%D 2D \ /
%D 2D \___________/
%D 2D
%D ren B0 A1 B2 A3 ==> \catB \catA \catB \catA
%D
%D (( B0 A1 -> .plabel= a R
%D B2 A3 -> .plabel= b R
%D A1 A3 ab! ab-curve^^
%D A1 B2 ab! ab-mid-id-R
%D B0 B2 ab! ab-curve__
%D ))
%D enddiagram
\pu
%D diagram cells-pure'-ne-sw-2
%D 2Dx 100 +25 +25 +25
%D 2D 100 __________
%D 2D / \
%D 2D / \
%D 2D R' / \
%D 2D 100 B0 ====> A1 ⇓id B2 ====> A3
%D 2D \ / R
%D 2D \ /
%D 2D \___________/
%D 2D
%D ren B0 A1 B2 A3 ==> \catB \catA \catB \catA
%D
%D (( B0 A1 => .plabel= a R'
%D B2 A3 => .plabel= b R'
%D A1 A3 ab! ab-curve^^
%D A1 B2 ab! ab-mid-id-R'
%D B0 B2 ab! ab-curve__
%D ))
%D enddiagram
\pu
% «defs-adjs-conj-mixed» (to ".defs-adjs-conj-mixed")
% The two adjunctions - the "mixed" diagrams:
%
% ___________ ___________
% / \ / \
% / ⇓ η' \ / \
% / =R'=> \ L / \ L
% A =====> B ⇓ τ A -----> B = A =====> B ⇓ σ A -----> B
% L' \ -R--> / L' \ /
% \ ⇓ ε / \ /
% \___________/ \___________/
%
% ___________ ___________
% / \ / \
% / ⇓ η \ / \
% R' / --L--> \ R' / \
% B =====> A ⇓ σ B -----> A = B =====> A ⇓ τ B -----> A
% \ =L'=> / R \ / R
% \ ⇓ ε' / \ /
% \___________/ \___________/
%
%
% ___________ ___________
% / \ / \
% / ⇓ η' \ / ⇓ η \
% / =R'=> \ / --L--> \
% A =====> B ⇓ τ A = A ⇓ σ B --R--> A
% L' --R--> ==L'=>
%
%
% R' --L--> ==R'=>
% B =====> A ⇓ σ B = B ⇓ τ A -----> B
% \ =L'=> / \ --R--> /
% \ ⇓ ε' / \ ⇓ ε /
% \___________/ \___________/
%
%
%D diagram cells-mixed-nw-se-before
%D 2Dx 100 +25 +25 +25
%D 2D 100 ___________
%D 2D / \
%D 2D / \
%D 2D / =R'=> \ L
%D 2D 100 A0 ====> B1 ⇓ σ A2 ----> B3
%D 2D L' \ --R--> /
%D 2D \ /
%D 2D \___________/
%D 2D
%D ren A0 B1 A2 B3 ==> \catA \catB \catA \catB
%D
%D (( A0 B1 => .plabel= b L'
%D A2 B3 -> .plabel= a L
%D A0 A2 ab! ab-eta'
%D B1 A2 ab! ab-tau
%D B1 B3 ab! ab-eps
%D ))
%D enddiagram
\pu
%D diagram cells-mixed-nw-se-after
%D 2Dx 100 +25 +25 +25
%D 2D 100 ___________
%D 2D / \
%D 2D / \
%D 2D / \ L
%D 2D 100 A0 ====> B1 ⇓ σ A2 ----> B3
%D 2D L' \ /
%D 2D \ /
%D 2D \___________/
%D 2D
%D ren A0 B1 A2 B3 ==> \catA \catB \catA \catB
%D
%D (( A0 B1 => .plabel= b L'
%D A2 B3 -> .plabel= a L
%D A0 A2 ab! ab-curve^^
%D B1 A2 ab! ab-name-mid: \Dn{σ}
%D B1 B3 ab! ab-curve__
%D ))
%D enddiagram
\pu
%D diagram cells-mixed-ne-sw-before
%D 2Dx 100 +25 +25 +25
%D 2D 100 ___________
%D 2D / \
%D 2D / ⇓ η \
%D 2D R' / -L--> \
%D 2D 100 B0 ====> A1 ⇓ σ B2 --R-> A3
%D 2D \ ==L'=> /
%D 2D \ /
%D 2D \___________/
%D 2D
%D ren B0 A1 B2 A3 ==> \catB \catA \catB \catA
%D
%D (( B0 A1 => .plabel= a R'
%D B2 A3 -> .plabel= b R
%D A1 A3 ab! ab-eta
%D A1 B2 ab! ab-sigma
%D B0 B2 ab! ab-eps'
%D ))
%D enddiagram
\pu
%D diagram cells-mixed-ne-sw-after
%D 2Dx 100 +25 +25 +25
%D 2D 100 ___________
%D 2D / \
%D 2D / ⇓ η \
%D 2D R' / -L--> \
%D 2D 100 B0 ====> A1 ⇓ σ B2 --R-> A3
%D 2D \ ==L'=> /
%D 2D \ /
%D 2D \___________/
%D 2D
%D ren B0 A1 B2 A3 ==> \catB \catA \catB \catA
%D
%D (( B0 A1 => .plabel= a R'
%D B2 A3 -> .plabel= b R
%D A1 A3 ab! ab-curve^^
%D A1 B2 ab! ab-name-mid: \Dn{τ}
%D B0 B2 ab! ab-curve__
%D ))
%D enddiagram
\pu
%D diagram cells-flip-eta-before
%D 2Dx 100 +25 +25
%D 2D 100 ___________
%D 2D / \
%D 2D / ⇓ η' \
%D 2D / =R'=> \
%D 2D 100 A0 ====> B1 ⇓ σ A2
%D 2D L' --R-->
%D 2D
%D ren A0 B1 A2 ==> \catA \catB \catA
%D
%D (( A0 B1 => .plabel= b L'
%D A0 A2 ab! ab-eta'
%D B1 A2 ab! ab-tau
%D ))
%D enddiagram
\pu
%D diagram cells-flip-eta-after
%D 2Dx 100 +25 +25
%D 2D 100 ___________
%D 2D / \
%D 2D / ⇓ η \
%D 2D / -L--> \
%D 2D 100 A0 ⇓ σ B1 --R-> A2
%D 2D =L'->
%D 2D
%D ren A0 B1 A2 ==> \catA \catB \catA
%D
%D (( A0 B1 ab! ab-sigma
%D A0 A2 ab! ab-eta
%D B1 A2 => .plabel= b R
%D ))
%D enddiagram
\pu
%D diagram cells-flip-eps-before
%D 2Dx 100 +25 +25
%D 2D
%D 2D R' --L-->
%D 2D 100 B0 ====> A1 ⇓ σ B2
%D 2D \ ==L'=> /
%D 2D \ ⇓ ε' /
%D 2D \___________/
%D 2D
%D ren B0 A1 B2 ==> \catB \catA \catB
%D
%D (( B0 A1 => .plabel= a R'
%D A1 B2 ab! ab-sigma
%D B0 B2 ab! ab-eps'
%D ))
%D enddiagram
\pu
%D diagram cells-flip-eps-after
%D 2Dx 100 +25 +25
%D 2D
%D 2D ==R'=> L
%D 2D 100 B0 ⇓ τ A1 -----> B2
%D 2D \ --R-> /
%D 2D \ ⇓ ε /
%D 2D \___________/
%D 2D
%D ren B0 A1 B2 ==> \catB \catA \catB
%D
%D (( A1 B2 -> .plabel= a L
%D B0 A1 ab! ab-tau
%D B0 B2 ab! ab-eps
%D ))
%D enddiagram
\pu
% «defs-adjs-conj-sigma-tau» (to ".defs-adjs-conj-sigma-tau")
% The cell diagrams for σ and τ:
%
%D diagram cells-sigma
%D 2Dx 100 +25
%D 2D --L-->
%D 2D 100 A0 ⇓ σ B1
%D 2D ==L'=>
%D 2D
%D ren A0 B1 ==> \catA \catB
%D
%D (( A0 B1 ab! ab-sigma
%D ))
%D enddiagram
\pu
%D diagram cells-tau
%D 2Dx 100 +25
%D 2D ==R'==>
%D 2D 100 B0 ⇓ τ A1
%D 2D ==L'=>
%D 2D
%D ren B0 A1 ==> \catB \catA
%D
%D (( B0 A1 ab! ab-tau
%D ))
%D enddiagram
\pu
% «defs-conj-cell-eqs» (to ".defs-conj-cell-eqs")
%
\sa {Conjungate adjunctions: cell equations (pure)} {
\begin{array}{rcl}
\diag{cells-pure-nw-se-1} &=&
\diag{cells-pure-nw-se-2} \\
\diag{cells-pure-ne-sw-1} &=&
\diag{cells-pure-ne-sw-2} \\\\
\diag{cells-pure'-nw-se-1} &=&
\diag{cells-pure'-nw-se-2} \\
\diag{cells-pure'-ne-sw-1} &=&
\diag{cells-pure'-ne-sw-2} \\
\end{array}
}
\sa {Conjungate adjunctions: cell equations (mixed)} {
\begin{array}{rcl}
\diag{cells-mixed-nw-se-before} &=&
\diag{cells-mixed-nw-se-after} \\
\diag{cells-mixed-ne-sw-before} &=&
\diag{cells-mixed-ne-sw-after} \\
\diag{cells-flip-eta-before} &=&
\diag{cells-flip-eta-after} \\
\diag{cells-flip-eps-before} &=&
\diag{cells-flip-eps-after} \\
\end{array}
}
\newpage
% ;-- adjunctions
% «adjunctions» (to ".adjunctions")
% (cwm2026p 2 "adjunctions")
% (cwm2026a "adjunctions")
% (find-books "__cats/__cats.el" "maclane" "80" "Definition. Let A and X be categories.")
% (find-symbolspage 70 "Table 139: Harpoons")
% (find-symbolstext 70 "Table 139: Harpoons")
\SLIDE{adjunctions}
\scalebox{0.4}{\def\colwidth{18cm}\firstcol{
$$
\begin{array}{ccc}
\ga{adjunction minipage CWM} &&
\ga{adjunction minipage Edrx} \\
\\
\\
\scalebox{2.0}{$\ga{adjunction diag CWM}$} &&
\scalebox{2.0}{$\ga{adjunction diag Edrx}$} \\
\end{array}
$$
}\anothercol{
}}
\newpage
% ;-- adjunctions-conj-pure
% «adjunctions-conj-pure» (to ".adjunctions-conj-pure")
% (cwm2026p 3 "adjunctions-conj-pure")
% (cwm2026a "adjunctions-conj-pure")
% (misp 51 "2-category-of-cats")
% (misa "2-category-of-cats")
\SLIDE{Conjugate adjunctions: cell equations (``pure'')}
\scalebox{0.6}{\def\colwidth{9.5cm}\firstcol{
$$\ga{Conjungate adjunctions: cell equations (pure)}
$$
}\anothercol{
\vspace*{2cm}
$$\begin{array}{rcl}
\bmat{η,L\\L,ε} &=& [L] \\\\
\bmat{R,η\\ε,R} &=& [R] \\\\
\bmat{η',L'\\L',ε'} &=& [L'] \\\\
\bmat{R',η'\\ε',R'} &=& [R'] \\\\
\end{array}
$$
}}
\newpage
% ;-- adjunctions-conj-mixed
% «adjunctions-conj-mixed» (to ".adjunctions-conj-mixed")
% (cwm2026p 4 "adjunctions-conj-mixed")
% (cwm2026a "adjunctions-conj-mixed")
\SLIDE{Conjugate adjunctions: cell equations (``mixed'')}
\scalebox{0.6}{\def\colwidth{9.5cm}\firstcol{
$$\ga{Conjungate adjunctions: cell equations (mixed)}
$$
}\anothercol{
\vspace*{2cm}
$$\begin{array}{rcl}
\bmat{η',L \\ L',τ,L \\ L',ε} &=& [σ] \\\\
\bmat{R',η \\ R',σ,R \\ ε',R} &=& [τ] \\\\
\bmat{η' \\ L',τ} &=& \bmat{η \\ σ,R} \\\\
\bmat{R',σ \\ ε'} &=& \bmat{τ,L \\ ε} \\\\
\end{array}
$$
}}
% (misp 51 "2-category-of-cats")
% (misa "2-category-of-cats")
% (find-riehlccpage (+ 18 44) "1.7. The 2-category of categories")
\newpage
% (misp 24 "internal-view-NT")
% (misa "internal-view-NT")
%D diagram generic-NT
%D 2Dx 100 +25
%D 2D 100 FA
%D 2D +10 A
%D 2D +10 GA
%D 2D
%D ren A FA GA ==> \ga{A} \ga{FA} \ga{GA}
%D
%D (( A FA |->
%D A GA |->
%D FA GA -> .plabel= r \ga{TA}
%D A FA GA midpoint |->
%D ))
%D enddiagram
\pu
\sa {sigma NT} {{
\sa {A} {A}
\sa {FA} {LA}
\sa {GA} {L'A}
\sa {TA} {σ_A}
\diag{generic-NT}
}}
\sa {tau NT} {{
\sa {A} {B}
\sa {FA} {R'B}
\sa {GA} {RB}
\sa {TA} {τ_B}
\diag{generic-NT}
}}
%D diagram sigma-triangle
%D 2Dx 100 +25
%D 2D 100 LA L'A
%D 2D
%D 2D +20 B
%D 2D
%D # ren ==>
%D
%D (( LA L'A -> .plabel= a σ_A
%D L'A B -> .plabel= r f
%D LA B -> .plabel= l f∘σ_A
%D ))
%D enddiagram
\pu
%D diagram tau-triangle
%D 2Dx 100 +25
%D 2D 100 A
%D 2D
%D 2D +20 R'B RB
%D 2D
%D # ren ==>
%D
%D (( A R'B -> .plabel= l g
%D R'B RB -> .plabel= b τ_B
%D A RB -> .plabel= r τ_B∘g
%D ))
%D enddiagram
\pu
\def\AoverB #1#2{\pmat{#1\\↓\\#2}}
\sa {L'A/B} {\AoverB{L'A}{B}}
\sa {LA/B} {\AoverB {LA}{B}}
\sa {A/R'B} {\AoverB{A}{R'B}}
\sa {A/RB} {\AoverB{A} {RB}}
%D diagram cwm-p100-6-edrx-1
%D 2Dx 100 +40
%D 2D 100 A0 <--- A1
%D 2D | |
%D 2D v v
%D 2D +40 A2 ---> A3
%D 2D
%D ren A0 A1 ==> \AoverB{L'A}{B} \AoverB{A}{R'B}
%D ren A2 A3 ==> \AoverB{LA}{B} \AoverB{A}{RB}
%D
%D (( A0 A1 <- .plabel= a ♭' sl^
%D A0 A1 -> .plabel= b ♯' sl_
%D A0 A2 -> .plabel= l ∘σ_A
%D A1 A3 -> .plabel= r τ_B∘
%D A2 A3 <- .plabel= a ♭ sl^
%D A2 A3 -> .plabel= b ♯ sl_
%D ))
%D enddiagram
\pu
%D diagram cwm-p100-6-edrx-2
%D 2Dx 100 +40
%D 2D 100 A0 <--- A1
%D 2D | |
%D 2D v v
%D 2D +40 A2 ---> A3
%D 2D
%D ren A0 A1 ==> \AoverB{L'R'B}{B} \AoverB{R'B}{R'B}
%D ren A2 A3 ==> \AoverB{LR'B}{B} \AoverB{R'B}{RB}
%D
%D (( A0 A1 <- .plabel= a ♭'
%D A0 A2 -> .plabel= l ∘σ_{R'B}
%D A1 A3 -> .plabel= r τ_B∘
%D A2 A3 -> .plabel= b ♯
%D ))
%D enddiagram
\pu
%D diagram cwm-p100-6-edrx-2-b
%D 2Dx 100 +80
%D 2D 100 A0 <--- A1
%D 2D | |
%D 2D +25 v A3'
%D 2D +10 A2 ---> A3
%D 2D
%D ren A0 A1 ==> ε'_B \id_{R'B}
%D ren A3' ==> τ_B=[τ]B
%D ren A2 A3 ==> \ga{cwm-p100-6-edrx-2-b-SW} \ga{cwm-p100-6-edrx-2-b-SE}
%D
%D (( A0 A1 <-| .plabel= a ♭'
%D A0 A2 |-> .plabel= l ∘σ_{R'B}
%D A1 A3' |-> .plabel= r τ_{B}∘
%D A2 A3 |-> .plabel= b ♯
%D ))
%D enddiagram
\pu
\sa {cwm-p100-6-edrx-2-b-SW} {{
\begin{array}[t]{l}
ε'_B∘σ_{R'B} \\
= [ε']B∘[R',σ]B \\
= \bsm{R',σ \\ ε'}B \\
\end{array}
}}
\sa {cwm-p100-6-edrx-2-b-SE} {{
\begin{array}[t]{l}
R(\bsm{R',σ \\ ε'}B) ∘ η_{RB} \\
= [\bsm{R',σ \\ ε'},R]B ∘ [R,η]B \\
= \bsm{R',σ,R \\ ε',R}B ∘ [R,η]B \\
= \bsm{R,η \\ R',σ,R \\ ε',R}B \\
\end{array}
}}
\scalebox{0.5}{\def\colwidth{9cm}\firstcol{
$$\begin{array}{cc}
\diag{cells-sigma} & \ga {sigma NT} \\
\diag{cells-tau} & \ga {tau NT} \\
\end{array}
$$
$$\diag{sigma-triangle}
\quad
\diag{cwm-p100-6-edrx-1}
\quad
\diag{tau-triangle}
$$
$$\diag{cwm-p100-6-edrx-2} \qquad
\diag{cwm-p100-6-edrx-2-b}
$$
}\anothercol{
}}
$$\pu
$$
% sigma and tau:
% (find-books "__cats/__cats.el" "maclane" "99" "7. Transformations of Adjoints")
% (find-books "__cats/__cats.el" "maclane" "100" "are said to be conjugate (for the given adjunctions) when the diagram")
% (find-es "diagxy" "diagxyto")
% (find-es "xypic" "two-and-three")
% (find-diagxypage 27 "\\two/<-< `<-< /^f_g")
% (find-diagxytext 27 "\\two/<-< `<-< /^f_g")
% ;-- writetoc
% «writetoc» (to ".writetoc")
\directlua{toclines:writetoc()}
% Writes in: (find-LATEXfile "2026cwm.mytoc")
% See: (to "toc")
% ;-- references
% «references» (to ".references")
%\printbibliography
% ;-- write-dnt-file
% «write-dnt-file» (to ".write-dnt-file")
% (find-fline "~/LATEX/" "2026cwm.dnt")
% (find-fline "~/LATEX/2026cwm.dnt")
%L write_dnt_file ("2026cwm.dnt")
\pu
\GenericWarning{Success:}{Success!!!} % Used by `M-x cv'
\end{document}
% (find-pdfpages2-links "~/LATEX/" "2026cwm")
% Local Variables:
% coding: utf-8-unix
% outline-regexp: "% +;--"
% ee-tla: "cwm"
% ee-tla: "cwm2026"
% End: