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