% (find-LATEX "2026what-is-yoneda.tex") % (defun c () (interactive) (find-LATEXsh "lualatex -record 2026what-is-yoneda.tex" :end)) % (defun C () (interactive) (find-LATEXsh "lualatex 2026what-is-yoneda.tex" "Success!!!")) % (defun D () (interactive) (find-pdf-page "~/LATEX/2026what-is-yoneda.pdf")) % (defun d () (interactive) (find-pdftools-page "~/LATEX/2026what-is-yoneda.pdf")) % (defun e () (interactive) (find-LATEX "2026what-is-yoneda.tex")) % (defun o () (interactive) (find-LATEX "2026what-is-yoneda.tex")) % (defun u () (interactive) (find-latex-upload-links "2026what-is-yoneda")) % (defun v () (interactive) (find-2a '(e) '(d))) % (defun d0 () (interactive) (find-ebuffer "2026what-is-yoneda.pdf")) % (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g)) % (defun oe () (interactive) (find-2a '(o) '(e))) % (code-eec-LATEX "2026what-is-yoneda") % (find-pdf-page "~/LATEX/2026what-is-yoneda.pdf") % (find-sh0 "cp -v ~/LATEX/2026what-is-yoneda.pdf /tmp/") % (find-sh0 "cp -v ~/LATEX/2026what-is-yoneda.pdf /tmp/pen/") % (find-xournalpp "/tmp/2026what-is-yoneda.pdf") % file:///home/edrx/LATEX/2026what-is-yoneda.pdf % file:///tmp/2026what-is-yoneda.pdf % file:///tmp/pen/2026what-is-yoneda.pdf % http://anggtwu.net/LATEX/2026what-is-yoneda.pdf % https://anggtwu.net/LATEX/2026what-is-yoneda.pdf % (find-LATEX "2019.mk") % (find-Deps1-links "Caepro5 Piecewise2 Maxima2") % (find-Deps1-cps "Caepro5 Piecewise2 Maxima2") % (find-Deps1-anggs "Caepro5 Piecewise2 Maxima2") % (find-MM-aula-links "2026what-is-yoneda" "2" "wiy" "wiy") % «.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") % «.defs-H-V-square» (to "defs-H-V-square") % % «.title» (to "title") % «.toc» (to "toc") % «.links» (to "links") % «.abstract» (to "abstract") % «.intro-1» (to "intro-1") % «.sfard» (to "sfard") % «.intro-2» (to "intro-2") % «.intro-3» (to "intro-3") % «.juxtaposition» (to "juxtaposition") % «.what-is-an-elephant» (to "what-is-an-elephant") % «.greer» (to "greer") % «.on-the-missing» (to "on-the-missing") % «.on-the-missing-2» (to "on-the-missing-2") % «.some-yonedas» (to "some-yonedas") % «.convention-TT» (to "convention-TT") % «.adjunctions» (to "adjunctions") % «.adjunction-with» (to "adjunction-with") % «.reconstruct-yoneda» (to "reconstruct-yoneda") % «.reconstruct-yoneda-2» (to "reconstruct-yoneda-2") % «.reconstruct-yoneda-3» (to "reconstruct-yoneda-3") % «.sierpinska» (to "sierpinska") % «.ausubel» (to "ausubel") % «.variants-of-ap» (to "variants-of-ap") % % «.writetoc» (to "writetoc") % «.references» (to "references") % «.make-with-bib» (to "make-with-bib") % ;-- 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 %L require "DiagTeX1" -- (find-angg "LUA/DiagTeX1.lua") \pu % «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)}} \long\def\ColorBlue #1{{\color{Blue1}#1}} \def\P#1{\left(#1\right)} \def\Nat{\mathsf{Nat}} \def\rarr{\ColorRed{⇒}} \def\univ{\mathsf{univ}} \def\Ring{\mathsf{Ring}} % «defs-H-V-square» (to ".defs-H-V-square") % H: a natural transformation, written as a horizontal arrow. % V: a morphism, written as a horizontal arrow. % (H)(V) = (square) % %D diagram DE %D 2Dx 100 %D 2D 100 D %D 2D | %D 2D v %D 2D +20 E %D 2D %D (( D E -> .plabel= l \ga{f} %D )) %D enddiagram \pu \sa {(DE)} {{\sa{f}{} \pdiag{DE}}} \sa {(DgE)} {{\sa{f}{f} \pdiag{DE}}} \sa {(FG)} {(F \to G)} \sa {(FTG)} {(F \ton{T} G)} \sa {(C->AR)} {((C{→}) \to (A{→}R-))} % Test: % %$$\ga{(FG)} % \ga{(FTG)} % \ga{(C->AR)} % \ga{(DE)} % \ga{(DgE)} %$$ %D diagram Color-NT-1 %D 2Dx 100 +40 %D 2D 100 A0 -> A1 %D 2D | | %D 2D | | %D 2D +20 v v %D 2D +7 A2 -> A3 %D 2D %D ren A0 A1 ==> \ColorRed{(C{→}D)} \ColorGreen{(A{→}RD)} %D ren A2 A3 ==> \ColorOrange{(C{→}E)} \ColorBlue{(A{→}RE)} %D %D (( A0 A1 -> %D A0 A2 -> %D A1 A3 -> %D A2 A3 -> %D )) %D enddiagram \pu %D diagram square-30 %D 2Dx 100 +30 %D 2D 100 \ga{nw} -> \ga{ne} %D 2D | | %D 2D v v %D 2D +20 \ga{sw} -> \ga{se} %D 2D %D (( \ga{nw} \ga{ne} -> .plabel= a \ga{n} %D \ga{nw} \ga{sw} -> .plabel= l \ga{w} %D \ga{ne} \ga{se} -> .plabel= r \ga{e} %D \ga{sw} \ga{se} -> .plabel= a \ga{s} %D )) %D enddiagram \pu %D diagram square-40 %D 2Dx 100 +40 %D 2D 100 \ga{nw} -> \ga{ne} %D 2D | | %D 2D v v %D 2D +20 \ga{sw} -> \ga{se} %D 2D %D (( \ga{nw} \ga{ne} -> .plabel= a \ga{n} %D \ga{nw} \ga{sw} -> .plabel= l \ga{w} %D \ga{ne} \ga{se} -> .plabel= r \ga{e} %D \ga{sw} \ga{se} -> .plabel= a \ga{s} %D )) %D enddiagram \pu \Sa{square-30}#1#2#3#4#5#6#7#8{{ \sa{nw}{#1} \sa{n}{#2} \sa{ne}{#3} \sa {w}{#4} \sa {e}{#5} \sa{sw}{#6} \sa{s}{#7} \sa{se}{#8} \diag{square-30} }} \Sa{square-40}#1#2#3#4#5#6#7#8{{ \sa{nw}{#1} \sa{n}{#2} \sa{ne}{#3} \sa {w}{#4} \sa {e}{#5} \sa{sw}{#6} \sa{s}{#7} \sa{se}{#8} \diag{square-40} }} \sa {(FTG)*(DgE)} {\P{\ga{square-30} {FD} {TD} {GD} {Fg} {Gg} {FE} {TE} {GE}}} \sa {(FG)*(DE)} {\P{\ga{square-30} {FD} {} {GD} {} {} {FE} {} {GE}}} \sa {(C->AR)*(DE)} {\P{\ga{square-40} {(C{→})D} {} {(A{→}R-)D} {} {} {(C{→})E} {} {(A{→}R-)E}}} % Test: %$$\begin{array}{rcc} % \ga{(FTG)} \ga{(DgE)} &=& \ga {(FTG)*(DgE)} \\\\[-5pt] % \ga{(FG)} \ga{(DE)} &=& \ga {(FG)*(DE)} \\\\[-5pt] % \ga{(FG)} \ga{(DE)} &=& \P{\diag{Color-NT-1}} \\\\[-5pt] % \ga{(C->AR)} \ga{(DE)} &=& \ga {(C->AR)*(DE)} \\ % \end{array} %$$ % ;-- title % _____ _ _ _ % |_ _(_) |_| | ___ _ __ __ _ __ _ ___ % | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \ % | | | | |_| | __/ | |_) | (_| | (_| | __/ % |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___| % |_| |___/ % % «title» (to ".title") % (wiyp 1 "title") % (wiya "title") \thispagestyle{empty} \begin{center} \vspace*{1.2cm} \begin{tabular}{c} {\Large {\bf What ``is'' the Yoneda Lemma?}} \\[4pt] \\[-9pt] %\ColorGray{(EBCT 2026)}\\[-5pt] \ColorGray{(EBCT 2026 - 2026 may 27)}\\[-5pt] \\[-9pt] % https://anggtwu.net/math-b.html#2026-wld % https://anggtwu.net/math-b.html#2026-ebct % https://anggtwu.net/2026-alguns-motivos-bel.html %\url{https://anggtwu.net/math-b.html\#2026-ebct} %\url{https://anggtwu.net/2026-alguns-motivos-bel.html} {\tiny\url{https://anggtwu.net/math-b.html\#2026-ebct}}\\[-8pt] {\tiny\url{https://anggtwu.net/2026-alguns-motivos-bel.html}}\\[8pt] % Eduardo Ochs \\ Psicopata do CEFET \\ RCN/PURO/UFF \end{tabular} \end{center} %\newpage % ;-- toc % «toc» (to ".toc") % (to "writetoc") \newpage % ;-- links % «links» (to ".links") % (wiyp 2 "links") % (wiya "links") %{\bf Links} % %\scalebox{0.6}{\def\colwidth{16cm}\firstcol{ %}\anothercol{ %}} \newpage % ;-- abstract % _ _ _ _ % / \ | |__ ___| |_ _ __ __ _ ___| |_ % / _ \ | '_ \/ __| __| '__/ _` |/ __| __| % / ___ \| |_) \__ \ |_| | | (_| | (__| |_ % /_/ \_\_.__/|___/\__|_| \__,_|\___|\__| % % «abstract» (to ".abstract") % (nilp 99 "abstract") % (nila "abstract") % (find-LATEXfile "2026ebct-abs.tex") \SLIDE{Abstract} \sa{width1}{12cm} \sa{width2}{8cm} \sa{width3}{6.5cm} \sa{width4}{3.5cm} \scalebox{0.52}{\def\colwidth{\ga{width1}}\firstcol{ {} The ``Yoneda Embedding'' is easy to remember, but to prove it we need to prove the ``Yoneda Lemma'' first; the ``Yoneda Lemma'' takes a lot of mental space. \msk Let me refer to that ``Yoneda Lemma'' as the ``Yoneda Lemma (in the strict sense)''; the ``Yoneda Lemma (in the wide sense)'' will be the ``Yoneda Lemma (in the strict sense)'' plus the ``Yoneda Embedding''. In the next paragraphs I will use the term``Yoneda Lemma'' to mean the ``Yoneda Lemma (in the wide sense)''. \msk What ``is'' the Yoneda Lemma? Let's call that question Q1, and let's split it into several subquestions in the usual way: \begin{itemize} \item[Q2] What is the best notation, and what are the best diagrams, for remembering the Yoneda Lemma after we understand it well? \item[Q3] What are the best motivating examples for the Yoneda Lemma? As an aside, what does ``best motivating examples'' mean? Can we formalize that idea? There is a partial answer in \cite{OchsMD}; are there other ones? \item[Q4] Imagine that you want to present the Yoneda Lemma for non-categorists, and imagine that you can prepare a video for your presentation, and complement it with a blog post. What are the figures and animations that you would choose? Which slogans would you choose? Why? \end{itemize} }\def\colwidth{\ga{width2}}\anothercol{ {} Note that I am sort of splitting ``what is the Yoneda Lemma?'' into several subquestions, and some of them are naturally associated to verbs: ``remember'', ``reconstruct'', ``generalize'', ``specialize'', ``present'', ``teach''; some others are nouns: ``analogies'', ``animations'', ``slogans''. \msk I only started studying Mathematical Education (``ME'') very recently, and I was very surprised to see that this division into subquestions is studied a lot in ME. One extreme example is the book \cite{Ma}, that discusses ``deep understanding'' in the context of teaching Arithmetic to children, but practically all the ideas in that book are trivial to adapt to Category Theory. A less extreme example is \cite{VanHiele}, that discusses ``levels of thinking'', mainly in the context of school geometry. \msk In my presentation I will show some texts and ideias from ME that I believe that would be especially useful to categorists. The ``recommended readings'' will be basically the references at the end of \cite{BadFoundations}. }} \newpage % ;-- intro-1 % «intro-1» (to ".intro-1") % (wiyp 99 "intro-1") % (wiya "intro-1") \SLIDE{Introduction} \scalebox{0.9}{\def\colwidth{9cm}\firstcol{ {} I was a \standout{very incompetent} Category Theorist. \ColorRed{Everybody understood the Yoneda Lemma -- except me}. \msk Why? What was going on? How could I diagnose the problem {\it in a way that would let me fix it?} \msk I knew that my {\sl working memory} -- see \cite{Greer} -- was relatively weak, and {\sl sometimes} I could understand concepts from CT by drawing lots of diagrams, but knowing that wasn't enough... \msk Then I started to study Mathematical Education to understand some difficulties of my students, and the lots of ideas fell into place. }\anothercol{ }} \newpage % ;-- sfard % ____ __ _ % / ___| / _| __ _ _ __ __| | % \___ \| |_ / _` | '__/ _` | % ___) | _| (_| | | | (_| | % |____/|_| \__,_|_| \__,_| % % «sfard» (to ".sfard") % (wiyp 4 "sfard") % (wiya "sfard") \SLIDE{Sfard} \bsk % (find-books "__analysis/__analysis.el" "sfard" "9" "2. The Quandary of Abstraction (and Transfer)") % (find-books "__analysis/__analysis.el" "sfard" "16" "3. The Quandary of Misconceptions") % (find-books "__analysis/__analysis.el" "sfard" "22" "4. The Quandary of Learning Disability") % (find-books "__analysis/__analysis.el" "sfard" "27" "5. The Quandary of Understanding") Some sections from \cite[pp.9--33]{Sfard}: (``Thinking as Communicating -- Human Development, the Growth of Discourses, and Mathematizing'') \msk \par I.1. The Quandary of Number \par I.2. The Quandary of Abstraction (and Transfer) \par I.3. The Quandary of Misconceptions \par I.4. The Quandary of Learning Disability \par I.5. The Quandary of Understanding \par I.6. Puzzling about Thinking - in a Nutshell % Yonedas: % (misp 10 "eta <-> alpha") % (misp 12 "CWM") % (misp 16 "eta <-> alpha com sanfone") % (misp 37 "eta <-> alpha com D->E") % (misp 42 "eta <-> alpha com TT") % (misp 47 "B(C,-) <-> Set(1,R-)") % (misp 48 "Y2 = B(C,-) <-> R") % (misp 49 "Y3 = ") % (misp 50 "Y5 = ") % (misp 51 "F-|U Ring Set") % (find-LATEX "2022on-the-missing.tex" "the-conventions" "(CYo)") \newpage % ;-- intro-2 % «intro-2» (to ".intro-2") % (wiyp 99 "intro-2") % (wiya "intro-2") \SLIDE{Introduction (2)} \scalebox{0.7}{\def\colwidth{15cm}\firstcol{ In short: mathematicians don't even know that Mathematical Education \standout{exists}. Here are some books, articles and chapters on ME: \begin{itemize} \item Anna Sfard: ``Thinking as Communicating -- Human Development, the Growth of Discourses, and Mathematizing'' \item Anna Sierpinska: ``Understanding in Mathematics'' \item Pierre van Hiele: ``Structure and Insight -- A Theory of Mathematics Education'' \item Dave Hewitt: ``Arbitrary and Necessary, part 2: Assisting Memory'' \item Drouhard/Teppo: ``Symbols and Language'' \end{itemize} They are canonical references (FSV of...), but even Eugenia Cheng and David Corfield don't cite them... My first plan for this talk was to start it by apologizing profusely and by saying ``people, my talk may look trivial but at least you will learn a bit about an area that mathematicians know nothing about''... }\anothercol{ }} \newpage % ;-- intro-3 % «intro-3» (to ".intro-3") % (wiyp 5 "intro-3") % (wiya "intro-3") \SLIDE{Introduction (3)} \scalebox{0.65}{\def\colwidth{14cm}\firstcol{ ...but then I found some nice technical things to present -- a family of operations \standout{with omitted names} that work like this, $$\begin{array}{rcc} \ga{(FTG)} \ga{(DgE)} &=& \ga {(FTG)*(DgE)} \\\\[-5pt] \ga{(FG)} \ga{(DE)} &=& \ga {(FG)*(DE)} \\\\[-5pt] \ga{(FG)} \ga{(DE)} &=& \P{\diag{Color-NT-1}} \\\\[-5pt] \ga{(C->AR)} \ga{(DE)} &=& \ga {(C->AR)*(DE)} \\ \end{array} $$ }\anothercol{ }} % (code-pdf-page "vanhiele" "~/books/__analysis/van_hiele__structure_and_insight_a_theory_of_mathematics_education.pdf") % (code-pdf-text "vanhiele" "~/books/__analysis/van_hiele__structure_and_insight_a_theory_of_mathematics_education.pdf" 10) \newpage % ;-- juxtaposition % «juxtaposition» (to ".juxtaposition") % (wiyp 99 "juxtaposition") % (wiya "juxtaposition") \SLIDE{Juxtapositions} \sa{width1}{9.5cm} \sa{width2}{7.5cm} \sa{width3}{6.5cm} \sa{width4}{3.5cm} \scalebox{1.0}{\def\colwidth{\ga{width1}}\firstcol{ {} From \cite[sec.\ 4.1]{BadFoundations}: \ssk In some cases, like $2(3+4)$, the juxtaposion means an operation that was elided, but whose name we know, and we can write it explicitly: $2·(3+4)$. In other cases the operation doesn't have a standard notation, and we have to improvise. Let's use the symbol `\standout{ap}' for application, and `\standout{s}' for substitute: % $$\begin{array}{rcl} 2(y+z) & \rarr & 2·(y+z) \\ f(y+z) & \rarr & f \; \standout{ap} \; (y+z) \\ (a+b)[a:=42] & \rarr & (a+b) \; \standout{s} \; [a:=42] \\ \end{array} $$ }\def\colwidth{\ga{width2}}\anothercol{ {} }\def\colwidth{\ga{width3}}\anothercol{ {} }\def\colwidth{\ga{width4}}\anothercol{ {} }} \newpage % ;-- variants-of-ap % «variants-of-ap» (to ".variants-of-ap") % (wiyp 99 "variants-of-ap") % (wiya "variants-of-ap") \SLIDE{Variants of \standout{ap}} $$\begin{array}{rcc} \ga{(FTG)} \ga{(DgE)} &=& \ga {(FTG)*(DgE)} \\\\[-5pt] \ga{(FG)} \ga{(DE)} &=& \ga {(FG)*(DE)} \\\\[-5pt] \end{array} $$ \newpage % ;-- what-is-an-elephant % _____ _ _ _ % | ____| | ___ _ __ | |__ __ _ _ __ | |_ % | _| | |/ _ \ '_ \| '_ \ / _` | '_ \| __| % | |___| | __/ |_) | | | | (_| | | | | |_ % |_____|_|\___| .__/|_| |_|\__,_|_| |_|\__| % |_| % % «what-is-an-elephant» (to ".what-is-an-elephant") % (wiyp 4 "what-is-an-elephant") % (wiya "what-is-an-elephant") \SLIDE{What is an elephant?} \sa{width1}{7cm} \sa{width2}{8cm} \sa{width3}{6.5cm} \sa{width4}{3.5cm} \scalebox{0.7}{\def\colwidth{\ga{width1}}\firstcol{ {} % (find-books "__analysis/__analysis.el" "bessis" "253" "simply the left front\\nfoot, or the trunk") % (find-includegraphics-links "/tmp/elephant.pdf") % (find-pdf-page "~/LATEX/2026what-is-yoneda/elephant.pdf") $$\includegraphics[width=7cm]{2026what-is-yoneda/elephant.pdf}$$ \bsk This is a \standout{drawing} of an elephant. }\def\colwidth{\ga{width2}}\anothercol{ {} From \cite[p.253]{Bessis}: \msk But that's not all. When a child sees a real elephant for the first time, even if they’ve never seen a picture or heard it spoken of before, if you point your finger at the elephant and say,``That's an elephant'' the child knows immediately what you’re talking about. \msk That's not as obvious as it may seem. What keeps the child from thinking that what you're calling an elephant is simply the left front foot, or the trunk, or a piece of the trunk, or a fly sitting on the trunk? \bsk \standout{What is the Yoneda Lemma?} \standout{What is a drawing of the Yoneda Lemma?} }\def\colwidth{\ga{width3}}\anothercol{ {} }\def\colwidth{\ga{width4}}\anothercol{ {} }} \newpage % ;-- greer % ____ % / ___|_ __ ___ ___ _ __ % | | _| '__/ _ \/ _ \ '__| % | |_| | | | __/ __/ | % \____|_| \___|\___|_| % % «greer» (to ".greer") % (wiyp 99 "greer") % (wiya "greer") \SLIDE{Reconstructive memory} \sa{width1}{6cm} \sa{width2}{6cm} \sa{width3}{6.5cm} \sa{width4}{3.5cm} \scalebox{0.9}{\def\colwidth{\ga{width1}}\firstcol{ {} % (find-books "__analysis/__analysis.el" "greer-cpmt" "22" "He stressed the reconstruaive") From \cite[p.23]{Greer}: \ssk He stressed the reconstructive nature of memory -- what is remembered is not an exact copy but a stripped-down version of the original. \bsk The figure at the right is from \cite[sec.1]{OchsIDARCT} -- it is how I remember the Frobenius Property... I forget the letters, and I remember a \standout{shape} and \standout{movement}. }\def\colwidth{\ga{width2}}\anothercol{ {} % (find-angg ".emacs" "idarct-preprint" "1" "1. Mental Space and Diagrams") $$\includegraphics[width=5cm]{frob-sketch-eps-converted-to}$$ }\def\colwidth{\ga{width3}}\anothercol{ {} }\def\colwidth{\ga{width4}}\anothercol{ {} }} \newpage % ;-- on-the-missing % «on-the-missing» (to ".on-the-missing") % (wiyp 99 "on-the-missing") % (wiya "on-the-missing") \SLIDE{On the the Missing Diagrams in Category Theory} % (misp 9 "the-conventions") % (misa "the-conventions") %D diagram Basic-Example %D 2Dx 100 +40 %D 2D 100 A1 %D 2D | %D 2D +20 A2 |-> A3 %D 2D %D 2D +15 B0 --> B1 %D 2D %D 2D +15 C0 --> C1 %D 2D \ | %D 2D +20 C2 %D 2D %D ren A1 ==> A %D ren A2 A3 ==> C RC %D ren B0 B1 ==> \catB \catA %D ren C0 C1 C2 ==> \catB(C,-) \catA(A,R-) ? %D %D (( A1 A3 -> .plabel= r η %D A2 A3 |-> %D %D B0 B1 -> .plabel= a R\phantom{mmm} %D %D C0 C1 -> .plabel= b α %D # C0 C2 -> .plabel= l \sm{ψ\\\text{(iso)}} %D # C1 C2 <-> %D %D C0 C1 midpoint A1 A3 midpoint <-> .curve= ^15pt %D )) %D enddiagram \pu \sa{width1}{8.5cm} \sa{width2}{7cm} \sa{width3}{6.5cm} \sa{width4}{3.5cm} \scalebox{0.65}{\def\colwidth{\ga{width1}}\firstcol{ {} % (misp 9 "the-conventions") % (misa "the-conventions") From \cite[sec.2]{OchsMD}: \ssk (CD): Our diagrams are made of components that are nodes and arrows. The nodes can contain arbitrary expressions. The arrows work as connectives, and each arrow can be interpreted as the top-level connective in the smallest subexpression that contains it. For example, the curved arrow in the diagram at the right can be interpreted as: % $$(A\ton{η}RC)↔(\catB(C,-) \ton{T} \catA(A,R-)). $$ (CAI): ``Above'' usually means ``inside'', or ``internal view''. In the diagram above the morphism $η:A→RC$ is in $\catA$ and $C$ is an object of $\catB$. Also, the arrow $C \mapsto RC$ is above $\catB \ton{R} \catA$, and this means that it is an internal view of the functor $R$. Note that {\sl usually} is not {\sl always} --- and $\catB \ton{R} \catA$ is not an internal view of $\catB(C,-) \ton{T} \catA(A,R-)$. }\def\colwidth{\ga{width2}}\anothercol{ {} $$\scalebox{1.8}{$ \diag{Basic-Example} $} $$ }\def\colwidth{\ga{width3}}\anothercol{ {} }\def\colwidth{\ga{width4}}\anothercol{ {} }} \newpage % ;-- on-the-missing-2 % «on-the-missing-2» (to ".on-the-missing-2") % (wiyp 99 "on-the-missing-2") % (wiya "on-the-missing-2") \SLIDE{More conventions} %D diagram yoneda-cwm-0-small %D 2Dx 100 +35 %D 2D 100 A1 %D 2D | %D 2D +20 A2 |-> A3 %D 2D %D 2D +10 B0 --> B1 %D 2D %D 2D +10 C0 --> C1 %D 2D \ | %D 2D +20 C2 %D 2D %D ren A1 ==> c %D ren A2 A3 ==> r Sr %D ren B0 B1 ==> D C %D ren C0 C1 C2 ==> D(r,-) C(c,S-) ? %D %D (( A1 A3 -> .plabel= r u %D A2 A3 |-> %D %D B0 B1 -> .plabel= a S\phantom{mmm} %D %D C0 C1 -> .plabel= b φ %D # C0 C2 -> .plabel= l \sm{ψ\\\text{(iso)}} %D # C1 C2 <-> %D %D C0 C1 midpoint A1 A3 midpoint <-> .curve= ^11pt %D )) %D enddiagram \pu \sa{width1}{6.5cm} \sa{width2}{7.5cm} \sa{width3}{6.5cm} \sa{width4}{3.5cm} \scalebox{0.8}{\def\colwidth{\ga{width1}}\firstcol{ {} % (misp 9 "the-conventions") % (misa "the-conventions") From \cite[sec.2]{OchsMD}: \ssk (COT): We use a notation as close to the original text as possible, especially when we are trying to draw the missing diagrams for some existing text. If we were drawing the missing diagrams for the Proposition 1 of \cite[Section III.2]{CWM2} our diagram would be the one at the right... \ssk ...but I hate Mac Lane's choice of letters, so I decided to use another notation here (i.e., in \cite{OchsMD}). \msk (CNSh): A translation of a diagram $D$ to another notation is drawn with the same shape as $D$. }\def\colwidth{\ga{width2}}\anothercol{ {} $$\scalebox{2.0}{$ \diag{yoneda-cwm-0-small} $} $$ }\def\colwidth{\ga{width3}}\anothercol{ {} }\def\colwidth{\ga{width4}}\anothercol{ {} }} \newpage % ;-- some-yonedas % «some-yonedas» (to ".some-yonedas") % (wiyp 4 "some-yonedas") % (wiya "some-yonedas") \SLIDE{Some Yonedas} \sa{width1}{7cm} \sa{width2}{12cm} \sa{width3}{6.5cm} \sa{width4}{3.5cm} \scalebox{0.6}{\def\colwidth{\ga{width1}}\firstcol{ {} % (find-books "__cats/__cats.el" "maclane" "59" "Proposition 1") % (find-books "__cats/__cats.el" "maclane" "61" "Lemma (Yoneda)") \par From \cite[p.61]{CWM2}: \ssk \par {\bf Lemma (Yoneda)}. \par If $K:D→\Set$ is a functor from $D$ \par and $r$ an object in $D$ \par (for $D$ a category with small hom-sets), \par there is a bijection \par $y: \Nat(D(r,-),K) ≅ Kr$ \par which sends each natural transformation $α:D(r,-) \tnto K$ to $α_r 1_r$, \par the image of the identity $r→r$. }\def\colwidth{\ga{width2}}\anothercol{ {} % (find-books "__cats/__cats.el" "smith-cat3" "351" "Theorem 187 (The Core Yoneda Lemma)") \par From \cite[p.351]{SmithCat3}: \ssk \par {\bf Theorem 187 (The Core Yoneda Lemma)}. \par For any object $A$ of the locally small category $\catC$, \par and any functor $F:\catC→\Set$, \par $\Nat(\catC(A,-),F) ≅ FA$. \bsk % (find-books "__cats/__cats.el" "smith-cat3" "356" "Theorem 190 (Full Yoneda Lemma)") \par From \cite[p.356]{SmithCat3}: \ssk \par {\bf Theorem 190 (Full Yoneda Lemma)}. \par For any locally small category $\catC$, object $A$ in $C$, \par and covariant functor $F : \catC → \Set$, \par $\Nat(\catC(A,-),F) ≅ FA$, \par both naturally in $A$ \par and naturally in $F$. }\def\colwidth{\ga{width3}}\anothercol{ {} }\def\colwidth{\ga{width4}}\anothercol{ {} }} \newpage % ;-- convention-TT % «convention-TT» (to ".convention-TT") % (wiyp 99 "convention-TT") % (wiya "convention-TT") \SLIDE{Convention: Type Theory} \sa{width1}{4.5cm} \sa{width2}{8.5cm} \sa{width3}{6.5cm} \sa{width4}{3.5cm} \scalebox{0.825}{\def\colwidth{\ga{width1}}\firstcol{ {} % (misp 9 "the-conventions") % (misa "the-conventions") From \cite[sec.2]{OchsMD}: \ssk (CTT): Our diagrams should be close to Type Theory: it should be possible to use them as a scaffolding for formalizing our text in (pseudocode for) a proof assistant. \msk \ColorRed{So the `$↓$' at the right is a hom-set, and the diagram would still makes sense if we deleted the $f$!} }\def\colwidth{\ga{width2}}\anothercol{ {} %D diagram Hom-set %D 2Dx 100 %D 2D 100 A %D 2D | %D 2D v %D 2D +20 B %D 2D +10 \catC %D 2D %D # ren ==> %D %D (( A B -> .plabel= l \ga{f} %D \catC place %D %D )) %D enddiagram %D \sa{diag with f}{\sa{f}{f} \diag{Hom-set}} \sa{diag without f}{\sa{f}{} \diag{Hom-set}} \sa{TT with f}{ \begin{tabular}{rcl} $\catC$ & is & a category \\ $A$ & $∈$ & $\Objs_\catC$ \\ $B$ & $∈$ & $\Objs_\catC$ \\ $A→B$ & is & $\Hom(A,B)$, i.e., \\ & & $\Hom_\catC(A,B)$ \\ $f$ & : & $A→B$, i.e., \\ $f$ & ∈ & $A→B$, i.e., \\ $f$ & ∈ & $\Hom_\catC(A,B)$ \\ \end{tabular} } \sa{TT without f}{ \begin{tabular}{rcl} $\catC$ & is & a category \\ $A$ & $∈$ & $\Objs_\catC$ \\ $B$ & $∈$ & $\Objs_\catC$ \\ $A→B$ & is & $\Hom(A,B)$, i.e., \\ & & $\Hom_\catC(A,B)$ \\ \end{tabular} } $$\pu \begin{array}{rcl} \scalebox{1.2}{$\ga{diag with f}$} &\quad& \ga{TT with f} \\\\[-2pt] \scalebox{1.2}{$\ga{diag without f}$} &\quad& \ga{TT without f} \\ \end{array} $$ }\def\colwidth{\ga{width3}}\anothercol{ {} }\def\colwidth{\ga{width4}}\anothercol{ {} }} \newpage % ;-- adjunctions % «adjunctions» (to ".adjunctions") % (wiyp 99 "adjunctions") % (wiya "adjunctions") \SLIDE{Two adjunctions:} $L⊣R$ and $(×B)⊣(B{→})$ % (misp 26 "adjunctions") % (misp 26 "internal-view-adjunction") % (misa "internal-view-adjunction") %D diagram generic-adjunction %D 2Dx 100 +25 %D 2D 100 LB <-| B %D 2D | | %D 2D v v %D 2D +20 C |--> RC %D 2D %D 2D +15 E <==> F %D 2D %D ren LB B ==> LA A %D ren C RC ==> B RB %D ren E F ==> \catB \catA %D %D (( LB B <-| # .plabel= a L_0 %D C RC |-> # .plabel= b R_0 %D %D LB RC harrownodes nil 20 nil <-> %D %D LB C -> # .plabel= l \sm{g^♭\\f} %D B RC -> # .plabel= r \sm{g\\f^♯} %D E F <- sl^ .plabel= a L %D E F -> sl_ .plabel= b R %D )) %D enddiagram \pu %D diagram prod-exp-adjunction %D 2Dx 100 +25 %D 2D 100 LB <-| B %D 2D | | %D 2D v v %D 2D +20 C |--> RC %D 2D %D 2D +15 E <==> F %D 2D %D ren LB B ==> A×B A %D ren C RC ==> C B{→}C %D ren E F ==> \Set \Set %D %D (( LB B <-| # .plabel= a L_0 %D C RC |-> # .plabel= b R_0 %D %D LB RC harrownodes nil 20 nil <-> %D %D LB C -> # .plabel= l \sm{g^♭\\f} %D B RC -> # .plabel= r \sm{g\\f^♯} %D E F <- sl^ .plabel= a (×B) %D E F -> sl_ .plabel= b (B{→}) %D )) %D enddiagram \pu \scalebox{1.5}{\def\colwidth{8cm}\firstcol{ $$\pu \diag{generic-adjunction} \qquad \diag{prod-exp-adjunction} $$ }\anothercol{ }} \newpage %-- adjunction-with % «adjunction-with» (to ".adjunction-with") % (wiyp 99 "adjunction-with") % (wiya "adjunction-with") \SLIDE{An adjunction with functions, units, etc} %D diagram generic-adjunction-with-with %D 2Dx 100 +20 +20 +20 +20 +20 %D 2D 100 D0 D1 %D 2D +20 B0 C0 D2 D3 E0 F0 %D 2D +20 B1 C1 D4 D5 E1 F1 %D 2D +20 D6 D7 %D 2D +15 H0 H1 %D 2D %D ren D0 D1 ==> LA' A' %D ren B0 C0 D2 D3 E0 F0 ==> LR LRB LA A A \id_\catA %D ren B1 C1 D4 D5 E1 F1 ==> \id_\catB B B RB RLA LR %D ren D6 D7 ==> B' RB' %D ren H0 H1 ==> \catB \catA %D %D (( B0 B1 -> .plabel= l ε %D C0 C1 -> .plabel= l ε_B %D %D D0 D1 <-| %D D0 D2 -> .plabel= l Lf %D D1 D3 -> .plabel= r f %D D0 D3 harrownodes nil 20 nil <-| %D D2 D3 <-| %D D2 D4 -> .plabel= l \sm{h^♭\\g} %D D3 D5 -> .plabel= r \sm{h\\g^♯} %D D2 D5 harrownodes nil 20 nil <-| sl^ %D D2 D5 harrownodes nil 20 nil |-> sl_ %D D4 D5 |-> %D D4 D6 -> .plabel= l k %D D5 D7 -> .plabel= r Rk %D D6 D7 |-> %D D4 D7 harrownodes nil 20 nil |-> %D %D E0 E1 -> .plabel= r η_A %D F0 F1 -> .plabel= r η %D %D H0 H1 <- sl^ .plabel= a L %D H0 H1 -> sl_ .plabel= b R %D )) %D enddiagram %D $$\pu \diag{generic-adjunction-with-with} $$ \newpage % ;-- reconstruct-yoneda % «reconstruct-yoneda» (to ".reconstruct-yoneda") % (wiyp 99 "reconstruct-yoneda") % (wiya "reconstruct-yoneda") \SLIDE{Reconstructing the functors in Yoneda} \sa{F}{?} \sa{G}{?} \sa{T}{?} \sa{f}{f} \sa{g}{g} \sa{eta}{η} %D diagram Color-NT-0 %D 2Dx 100 +30 %D 2D 100 A1 %D 2D | %D 2D +20 A2 |-> A3 %D 2D | | %D 2D +20 A4 |-> A5 %D 2D | | %D 2D +20 A6 |-> A7 %D 2D %D 2D +15 B0 --> B1 %D 2D %D ren A1 ==> A %D ren A2 A3 ==> C RC %D ren A4 A5 ==> D RD %D ren A6 A7 ==> E RE %D ren B0 B1 ==> \ga{F} \ga{G} %D %D (( A1 A3 -> .plabel= r \ga{eta} %D A2 A3 |-> %D A2 A4 -> .plabel= l \ga{f} .arrowColor= Red %D A3 A5 -> # .plabel= r Rf %D A2 A5 harrownodes nil 20 nil |-> %D A4 A5 |-> %D A4 A6 -> .plabel= l \ga{g} %D A5 A7 -> # .plabel= r Rg %D A4 A7 harrownodes nil 20 nil |-> %D A6 A7 |-> %D A1 A5 -> .slide= 15pt .arrowColor= Green %D A1 A7 -> .slide= 25pt .arrowColor= Blue %D %D A2 A6 -> .slide= -10pt .arrowColor= Orange %D %D B0 B1 -> .plabel= a \ga{T} %D )) %D enddiagram \pu %D diagram Color-NT-1 %D 2Dx 100 +40 %D 2D 100 A0 -> A1 %D 2D | | %D 2D | | %D 2D +20 v v %D 2D +7 A2 -> A3 %D 2D %D ren A0 A1 ==> \ColorRed{(C{→}D)} \ColorGreen{(A{→}RD)} %D ren A2 A3 ==> \ColorOrange{(C{→}E)} \ColorBlue{(A{→}RE)} %D %D (( A0 A1 -> %D A0 A2 -> %D A1 A3 -> %D A2 A3 -> %D )) %D enddiagram \pu %D diagram Color-NT-2 %D 2Dx 100 +35 %D 2D 100 B0 -> B1 %D 2D | | %D 2D | v %D 2D +20 v B3 %D 2D +7 B2 -> B3' %D 2D %D ren B0 B1 ==> \ColorRed{f} \ColorGreen{η;Rf} %D ren B3 ==> \ColorBlue{(η;Rf);Rg} %D ren B2 B3' ==> \ColorOrange{f;g} \ColorBlue{η;R(f;g)} %D %D (( B0 B1 |-> %D B0 B2 |-> %D B1 B3 |-> %D B2 B3' |-> %D )) %D enddiagram \pu \scalebox{0.95}{\def\colwidth{13cm}\firstcol{ $$\diag{Color-NT-0} \qquad \qquad \diag{Color-NT-1} \qquad \diag{Color-NT-2} $$ }\anothercol{ }} \newpage % ;-- reconstruct-yoneda-2 % «reconstruct-yoneda-2» (to ".reconstruct-yoneda-2") % (wiyp 99 "reconstruct-yoneda-2") % (wiya "reconstruct-yoneda-2") \SLIDE{Reconstructing the functors in Yoneda (2)} \sa{F}{F} \sa{G}{G} \sa{T}{} \sa{f}{} \sa{g}{} \sa{eta}{} \scalebox{0.9}{\def\colwidth{13cm}\firstcol{ $$\diag{Color-NT-0} \qquad \qquad \begin{array}{rcl} \ga{(FG)}\ga{(DE)} &=& \P{\diag{Color-NT-1}} \\ \end{array} $$ }\anothercol{ }} \newpage %-- reconstruct-yoneda-3 % «reconstruct-yoneda-3» (to ".reconstruct-yoneda-3") % (wiyp 99 "reconstruct-yoneda-3") % (wiya "reconstruct-yoneda-3") \SLIDE{Reconstructing the functors in Yoneda (3)} \scalebox{0.8}{\def\colwidth{13cm}\firstcol{ $$\diag{Color-NT-0} \qquad \qquad \begin{array}{rcl} \ga{(FG)} \ga{(DE)} &=& \P{\diag{Color-NT-1}} \\\\[-5pt] \ga{(C->AR)} \ga{(DE)} &=& \ga{(C->AR)*(DE)} \\ \end{array} $$ }\anothercol{ }} \newpage % ;-- ausubel % _ _ _ % / \ _ _ ___ _ _| |__ ___| | % / _ \| | | / __| | | | '_ \ / _ \ | % / ___ \ |_| \__ \ |_| | |_) | __/ | % /_/ \_\__,_|___/\__,_|_.__/ \___|_| % % «ausubel» (to ".ausubel") % (wiyp 99 "ausubel") % (wiya "ausubel") \SLIDE{Ausubel} \sa{width1}{13cm} \sa{width2}{7.5cm} \sa{width3}{6.5cm} \sa{width4}{3.5cm} \scalebox{0.8}{\def\colwidth{\ga{width1}}\firstcol{ {} % (find-books "__analysis/__analysis.el" "ausubel-novak" "16" "Although knowledge of causation does not imply") From: \cite[p.16]{AusubelEP} \msk {\bf The Interdependence of Theories of Learning and Theories of Teaching} \ssk Although knowledge of causation does not imply immediate discovery of control procedures, it aids in the search for such procedures. For one thing, it narrows the field; for another, it enables one to try procedures that have proved successful in controlling related conditions. Knowing that tuberculosis was caused by a microorganism, for example, did not immediately provide us with a cure or a preventative. But it enabled us to try approaches--such as vaccines, immune sera, antisepsis, quarantine, and chemotherapy--that had been used successfully in treating other infectious diseases. In the same sense, knowledge of the cause of cancer would help immeasurably in discovering a cure, and knowledge of the nature and relevant variables involved in concept acquisition would be of invaluable assistance in devising effective methods of teaching concepts. }\def\colwidth{\ga{width2}}\anothercol{ {} }\def\colwidth{\ga{width3}}\anothercol{ {} }\def\colwidth{\ga{width4}}\anothercol{ {} }} \newpage % ;-- writetoc % «writetoc» (to ".writetoc") \directlua{toclines:writetoc()} % Writes in: (find-LATEXfile "2026what-is-yoneda.mytoc") % See: (to "toc") % ;-- references % «references» (to ".references") \printbibliography % ;-- write-dnt-file % «write-dnt-file» (to ".write-dnt-file") % (find-fline "~/LATEX/" "2026what-is-yoneda.dnt") % (find-fline "~/LATEX/2026what-is-yoneda.dnt") %L write_dnt_file ("2026what-is-yoneda.dnt") \pu \GenericWarning{Success:}{Success!!!} % Used by `M-x cv' \end{document} % (find-pdfpages2-links "~/LATEX/" "2026what-is-yoneda") % ;-- make-with-bib % «make-with-bib» (to ".make-with-bib") % (misa "make-arxiv")  (eepitch-shell)  (eepitch-kill)  (eepitch-shell) cd ~/LATEX/ # which biber # biber --version make -f 2019.mk STEM=2026what-is-yoneda veryclean lualatex 2026what-is-yoneda.tex biber 2026what-is-yoneda lualatex 2026what-is-yoneda.tex # (find-pdf-page "~/LATEX/2026what-is-yoneda.pdf") % Local Variables: % coding: utf-8-unix % outline-regexp: "% +;--" % ee-tla: "wiy" % ee-tla: "wiy" % End: