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\title{Kind of proposal (Talk)\\ %Inform if it is a talk or a poster
      \textbf{What ``is'' the Yoneda Lemma?} 
}

\author{
	{\large  \textsc{Eduardo Ochs}}\thanks{eduardoochs@gmail.com}\\
	{\small PURO/UFF, Rio das Ostras, RJ, Brasil} \\
}

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%-------------------------------- KEYWORDS -----------------------------------%

\textbf{Keywords:} Diagrammatic Reasoning, Mathematical Education

\vspace{1cm}

%-------------------------------- ABSTRACT ------------------------------------%

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.

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)''.

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}

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''.

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.

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}.



%------------------------------- REFERENCES -----------------------------------%

\begin{thebibliography}{00}

% Proceedings (How to cite: \cite{Ayla-Rinconetal-2016})

\bibitem{OchsMD} E. Ochs. {\it On the Missing Diagrams in Category
    Theory (First-Person Version)}.
  {\tt https://anggtwu.net/math-b.html\#2022-md}. 2022. arXiv:
  2401.12345.

\bibitem{Ma} L. Ma. {\it Knowing and Teaching Elementary Mathematics
    (Anniversary Edition)}. Routledge, 2010.

\bibitem{VanHiele} P. M. van Hiele. {\it Structure and Insight - A
    Theory of Mathematics Education}. Academic Press, 1986.

\bibitem{BadFoundations} E. Ochs. {\it Bad Foundations and Manipulable
    Objects}. {\tt
    https://anggtwu.net/math-b.html\#2026-bad-foundations}. 2026.
  arXiv: 2602.16026.

\end{thebibliography}

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