% (find-LATEX "2026riehl.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2026riehl.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2026riehl.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2026riehl.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2026riehl.pdf"))
% (defun e () (interactive) (find-LATEX "2026riehl.tex"))
% (defun o () (interactive) (find-LATEX "2020riehl.tex"))
% (defun u () (interactive) (find-latex-upload-links "2026riehl"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2026riehl.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (defun oe () (interactive) (find-2a '(o) '(e)))
%          (code-eec-LATEX "2026riehl")
% (find-pdf-page   "~/LATEX/2026riehl.pdf")
% (find-sh0 "cp -v  ~/LATEX/2026riehl.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2026riehl.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2026riehl.pdf")
%   file:///home/edrx/LATEX/2026riehl.pdf
%               file:///tmp/2026riehl.pdf
%           file:///tmp/pen/2026riehl.pdf
%  http://anggtwu.net/LATEX/2026riehl.pdf
% https://anggtwu.net/LATEX/2026riehl.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-cps   "Caepro5 Piecewise2 Maxima2 DiagForth1 DiagTeX1")
% (find-Deps1-anggs "Caepro5 Piecewise2 Maxima2")
% (find-MM-aula-links "2026riehl" "2" "rie2026" "rie")

% «.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-abcells»		(to "defs-abcells")
% «.defs-p46»			(to "defs-p46")
% «.horiz-and-vert»		(to "horiz-and-vert")
% «.horizontal-comp»		(to "horizontal-comp")
%
% «.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")
%L require "DiagTeX1"    -- (find-angg "LUA/DiagTeX1.lua")
\pu



% ;-- title
%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
% (rie2026p 1 "title")
% (rie2026a   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Notes on Emily Riehl's book}

{\bf \Large (``Categories in Context'')}

\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")
% (rie2026p 2 "links")
% (rie2026a   "links")

%{\bf Links}
%
%\scalebox{0.6}{\def\colwidth{16cm}\firstcol{
%}\anothercol{
%}}



% (find-books "__cats/__cats.el" "riehl" "46" "appearing in Lemma 1.7.4 are defined by whiskering")

% «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="]("^24pt") end
%L forths[".curve^"]  = function () forths[".curve="]( "^8pt") end
%L forths[".curve_"]  = function () forths[".curve="]( "_8pt") end
%L forths[".curve__"] = function () forths[".curve="]("_24pt") 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 = -7
%L y_eps =  7
%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-curve^^"] = function () dxyrun "ab@ -> .curve^^" end
%L forths["ab-curve^"]  = function () dxyrun "ab@ -> .curve^"  end
%L forths["ab-curve_"]  = function () dxyrun "ab@ -> .curve_"  end
%L forths["ab-curve__"] = function () dxyrun "ab@ -> .curve__" end

\def\Dn#1{{\scriptstyle \Downarrow \, #1}}

% «defs-p46»  (to ".defs-p46")

%D diagram p46-1
%D 2Dx     100      +25      +25      +25
%D 2D                   H
%D 2D          F      ----->     L
%D 2D  100 C -----> D  ⇓ β   E -----> F
%D 2D                 ----->
%D 2D                   K
%D 2D
%D ren C D E F ==> \catC \catD \catE \catF
%D
%D (( C D -> .plabel= a F
%D    D E -> .plabel= a H .curve^
%D    D E ab!  ab-name-mid: \Dn{β}
%D    D E -> .plabel= b K .curve_
%D    E F -> .plabel= a L
%D ))
%D enddiagram
\pu

%D diagram p46-2
%D 2Dx     100      +25      +25
%D 2D          F        J   
%D 2D        ----->   ----->
%D 2D         ⇓ α      ⇓ α  
%D 2D  100 C --G--> D --K--> E
%D 2D         ⇓ β      ⇓ β  
%D 2D        ----->   ----->
%D 2D          H        L   
%D 2D
%D ren C D E ==> \catC \catD \catE
%D
%D (( C D -> .plabel= a F .curve^^
%D    C D -> .plabel= m G
%D    C D -> .plabel= b H .curve__
%D    D E -> .plabel= a J .curve^^
%D    D E -> .plabel= m K
%D    D E -> .plabel= b L .curve__
%D    C D ab!  ab-name-up:   \Dn{α}
%D    C D ab!  ab-name-down: \Dn{β}
%D    D E ab!  ab-name-up:   \Dn{γ}
%D    D E ab!  ab-name-down: \Dn{δ}
%D ))
%D enddiagram
\pu

%D diagram p46-3
%D 2Dx     100      +25      +25
%D 2D          F        J   
%D 2D        ----->   ----->
%D 2D  100 C  ⇓β·α  D  ⇓δ·γ  E
%D 2D        ----->   ----->
%D 2D          H        L   
%D 2D
%D ren C D E ==> \catC \catD \catE
%D
%D (( C D -> .plabel= a F .curve^^
%D    C D -> .plabel= b H .curve__
%D    D E -> .plabel= a J .curve^^
%D    D E -> .plabel= b L .curve__
%D    C D ab!  ab-name-mid: \Dn{\ga{bb.aa}}
%D    D E ab!  ab-name-mid: \Dn{\ga{dd.cc}}
%D ))
%D enddiagram
\pu

%D diagram p46-4
%D 2Dx     100      +25      +25
%D 2D               JF   
%D 2D        -------------->
%D 2D             ⇓ γ*α
%D 2D  100 C -------KG-----> E
%D 2D             ⇓ γ*β  
%D 2D        -------------->
%D 2D              LH   
%D 2D
%D ren C E ==> \catC \catE
%D
%D (( C E -> .plabel= a \ga{JF} .curve^^
%D    C E -> .plabel= m \ga{KG}
%D    C E -> .plabel= b \ga{LH} .curve__
%D    C E ab!  ab-name-up:   \Dn{\ga{cc*aa}}
%D    C E ab!  ab-name-down: \Dn{\ga{dd*bb}}
%D ))
%D enddiagram
\pu

\sa {p46-3 Riehl} {{
  \sa   {bb.aa} {β·α}
  \sa   {dd.cc} {δ·γ}
  \diag {p46-3}
}}
\sa {p46-3 Edrx} {{
  \sa   {bb.aa} {\bsm{α\\β}}
  \sa   {dd.cc} {\bsm{γ\\δ}}
  \diag {p46-3}
}}

\sa {p46-4 Riehl} {{
  \sa   {JF} {JF}
  \sa   {KG} {KG}
  \sa   {LH} {LH}
  \sa   {cc*aa} {γ*α}
  \sa   {dd*bb} {δ*β}
  \diag {p46-4}
}}
\sa {p46-4 Edrx ()} {{
  \sa   {JF} {(F,J)}
  \sa   {KG} {(G,K)}
  \sa   {LH} {(H,L)}
  \sa   {cc*aa} {[α,γ]}
  \sa   {dd*bb} {[β,δ]}
  \diag {p46-4}
}}
\sa {p46-4 Edrx} {{
  \sa   {JF} {F,J}
  \sa   {KG} {G,K}
  \sa   {LH} {H,L}
  \sa   {cc*aa} {α,γ}
  \sa   {dd*bb} {β,δ}
  \diag {p46-4}
}}

\newpage
% ;-- horiz-and-vert
% «horiz-and-vert»  (to ".horiz-and-vert")
% (rie2026p 2 "horiz-and-vert")
% (rie2026a   "horiz-and-vert")

\SLIDE{Horizontal and vertical compositions}


\scalebox{0.65}{\def\colwidth{15.5cm}\firstcol{

From [Riehl, p.46]...

In my notation:

an outer $[·]$ stresses that the result is a NT, and

an outer $(·)$ stresses that the result is a functor.
%
$$\begin{array}{ccc}
  \diag{p46-2} \\
  \ga {p46-3 Riehl} = \ga {p46-4 Riehl}
    & (δ·γ)*(β·α)   = (δ*β)·(γ*α) \\
  \ga {p46-3 Edrx}  = \ga {p46-4 Edrx}
    &  \bmat{α\\β},\bmat{γ\\δ}  = \bmat{ α,γ \\{} β,δ }
       : (F,J) \tnto (H,L) \\
  \end{array}
$$

}\anothercol{
}}


\newpage




\scalebox{0.6}{\def\colwidth{9cm}\firstcol{

$$\begin{array}{ccc}
  \diag{p46-1} & LβF : LHF ⇒ LKF \\
               & [F,β,L] : (F,H,L) \tnto (F,K,L) \\
  %
  \diag{p46-2} \\
  \ga {p46-3 Riehl} = \ga {p46-4 Riehl}
    & (δ·γ)*(β·α)   = (δ*β)·(γ*α) \\
  \ga {p46-3 Edrx}  = \ga {p46-4 Edrx ()}
    & [\bmat{α\\β},\bmat{γ\\δ}] = \bmat{[α,γ]\\{}[β,δ]} \\
  \ga {p46-3 Edrx}  = \ga {p46-4 Edrx}
    &  \bmat{α\\β},\bmat{γ\\δ}  = \bmat{ α,γ \\{} β,δ } \\
  \end{array}
$$

}\anothercol{
}}


\newpage
% ;-- horizontal-comp
% «horizontal-comp»  (to ".horizontal-comp")
% (rie2026p 5 "horizontal-comp")
% (rie2026a    "horizontal-comp")

\SLIDE{Horizontal composition}

%L -- (find-angg "LUA/DiagTeX1.lua" "arrowColor")
%L forthe[".arrowcolorga="] = "w"
%L forths[".arrowcolorga="] = function (colortag)
%L    local colorga = format("\\colorga{%s}", colortag)
%L    local fmt = "{{"..colorga.."%s}}%%"
%L    local fmt = "{"..colorga.."%s}%%"
%L    local fmt = "{"..colorga.."%s\\color{black}}%%"
%L    forths[".arrowcolorwrap="](fmt)
%L   end
\pu
\def\colorga#1{\color{\ifgaundefined{#1}black\else\ga{#1}\fi}}




%D diagram sigma-tau-prism
%D 2Dx     100 +25  +30 +20
%D 2D  100          HFA
%D 2D  +10     FA    | \
%D 2D  +10   /  |    |  KFA
%D 2D  +15 A    |    |   |
%D 2D  +15   \  |   HGA  |
%D 2D  +10     GA      \ |
%D 2D  +10              KGA
%D 2D
%D (( A  FA |->  .arrowcolorga= A->FA
%D    A  GA |->  .arrowcolorga= A->GA
%D    FA HFA |-> .arrowcolorga= FA->HFA
%D    FA KFA |-> .arrowcolorga= FA->KFA
%D    GA HGA |-> .arrowcolorga= GA->HGA
%D    GA KGA |-> .arrowcolorga= KA->KGA
%D     FA  GA -> .plabel= m  σ_A
%D    HFA HGA -> .plabel= m Hσ_A
%D    KFA KGA -> .plabel= m Kσ_A
%D    HFA KFA -> .plabel= m τ_{FA}
%D    HGA KGA -> .plabel= m τ_{GA}
%D ))
%D enddiagram
\pu

%D diagram F,tau/sigma,K-open
%D 2Dx     100 +20 +30 +0
%D 2D  100         HFA
%D 2D  +20 A   FA  KFA
%D 2D  +20     GA  KGA
%D 2D
%D (( A  FA  |-> .arrowcolorga= A->FA  
%D    A  GA  |-> .arrowcolorga= A->GA  
%D    FA HFA |-> .arrowcolorga= FA->HFA
%D    FA KFA |-> .arrowcolorga= FA->KFA
%D  # GA HGA |-> .arrowcolorga= GA->HGA
%D    GA KGA |-> .arrowcolorga= KA->KGA
%D     FA  GA -> .plabel= r σ_A
%D  # HFA HGA -> .plabel= r Hσ_A
%D    KFA KGA -> .plabel= r Kσ_A
%D    HFA KFA -> .plabel= r τ_{FA}
%D  # HGA KGA -> .plabel= r τ_{GA}
%D ))
%D enddiagram
\pu

%D diagram sigma,F/G,tau-open
%D 2Dx     100 +20 +30 +0
%D 2D  100     FA  HFA
%D 2D  +20 A   GA  HGA
%D 2D  +20         KGA
%D 2D
%D (( A  FA  |-> .arrowcolorga= A->FA
%D    A  GA  |-> .arrowcolorga= A->GA
%D    FA HFA |-> .arrowcolorga= FA->HFA
%D  # FA KFA |-> .arrowcolorga= FA->KFA
%D    GA HGA |-> .arrowcolorga= GA->HGA
%D    GA KGA |-> .arrowcolorga= KA->KGA
%D     FA  GA -> .plabel= r σ_A
%D    HFA HGA -> .plabel= r Hσ_A
%D  # KFA KGA -> .plabel= r Kσ_A
%D  # HFA KFA -> .plabel= r τ_{FA}
%D    HGA KGA -> .plabel= r τ_{GA}
%D ))
%D enddiagram
\pu

%D diagram sigma,tau-cells
%D 2Dx     100   +25   +25
%D 2D  100 \catA \catB \catC
%D 2D
%D (( \catA \catB -> .curve^ .plabel= a F .arrowcolorga= A->FA
%D    \catA \catB -> .curve_ .plabel= b G .arrowcolorga= A->GA
%D    \catB \catC -> .curve^ .plabel= a H .arrowcolorga= B->HB
%D    \catB \catC -> .curve_ .plabel= b K .arrowcolorga= B->KB
%D    \catA \catB ab! ab-name-mid: \Dn{σ}
%D    \catB \catC ab! ab-name-mid: \Dn{τ}
%D ))
%D enddiagram
\pu

%D diagram F,tau/sigma,K-cells
%D 2Dx     100   +25   +25
%D 2D  100 \catA \catB \catC
%D 2D
%D (( \catA \catB ->          .plabel= a F .arrowcolorga= A->FA
%D    \catA \catB -> .curve__ .plabel= b G .arrowcolorga= A->GA
%D    \catB \catC -> .curve^^ .plabel= a H .arrowcolorga= B->HB
%D    \catB \catC ->          .plabel= b K .arrowcolorga= B->KB
%D    \catA \catB ab! ab-name-down: \Dn{σ}
%D    \catB \catC ab! ab-name-up:   \Dn{τ}
%D ))
%D enddiagram
\pu

%D diagram sigma,F/G,tau-cells
%D 2Dx     100   +25   +25
%D 2D  100 \catA \catB \catC
%D 2D
%D (( \catA \catB -> .curve^^ .plabel= a F .arrowcolorga= A->FA
%D    \catA \catB ->          .plabel= b G .arrowcolorga= A->GA
%D    \catB \catC ->          .plabel= a H .arrowcolorga= B->HB
%D    \catB \catC -> .curve__ .plabel= b K .arrowcolorga= B->KB
%D    \catA \catB ab! ab-name-up: \Dn{σ}
%D    \catB \catC ab! ab-name-down: \Dn{τ}
%D ))
%D enddiagram
\pu


\scalebox{0.5}{\def\colwidth{9cm}\firstcol{

\sa {A->FA} {orange}

$$\diag{sigma,tau-cells}
  \qquad
  \diag{sigma-tau-prism}
$$

$$\diag{F,tau/sigma,K-cells}
  \qquad
  \diag{F,tau/sigma,K-open}
$$

$$\diag{sigma,F/G,tau-cells}
  \qquad
  \diag{sigma,F/G,tau-open}
$$

}\anothercol{
}}









% ;-- writetoc
% «writetoc»  (to ".writetoc")
\directlua{toclines:writetoc()}
% Writes in: (find-LATEXfile "2026riehl.mytoc")
% See: (to "toc")

% ;-- references
% «references»  (to ".references")
%\printbibliography

% ;-- write-dnt-file
% «write-dnt-file»  (to ".write-dnt-file")
% (find-fline "~/LATEX/" "2026riehl.dnt")
% (find-fline    "~/LATEX/2026riehl.dnt")
%L write_dnt_file       ("2026riehl.dnt")
\pu


\GenericWarning{Success:}{Success!!!}  % Used by `M-x cv'

\end{document}

% (find-pdfpages2-links "~/LATEX/" "2026riehl")


% Local Variables:
% coding: utf-8-unix
% outline-regexp: "% +;--"
% ee-tla: "rie"
% ee-tla: "rie2026"
% End: