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\begin{document}
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\begin{abstract}
\textbf{Note to the organizers of the ACT2019:} I am submitting this
stub-like-ish abstract to try to make the system happy, or at least
aware of my existence... this, ahem, 𝐢{thing}, contains most of the
diagrams that I intend the final version to have, but the text is a
mess, and I won't be able to fix it until the end of May 03 AoE... I
will submit a final version in two more days if it can be accepted
with penalty points.
I am really embarassed. I am the author of the package (``dednat6'')
that produces the diagrams that appear here... it 𝐢{used} to depend
on LuaLaTeX, but the compositionality class is incompatible with
LuaLaTeX, and the task of creating workarounds for this took me FAR
longer than I predicted. My fault. ${=}($
\textbf{Proto-abstract} (may change):
Different people have different ways of remembering theorems. A
person with a very visual mind may remember a theorem in Category
Theory mainly by the shape of a diagram and the order in which its
objects are constructed. For such a person most books on Category
Theory feel as if they have lots of missing diagrams, that she has
to reconstruct if she wants to understand the subject.
The shape of a categorical diagram remains the same if we specialize
it to a particular case --- and this means that we can sometimes
remember a general diagram, and the theorems associated to it, from
the diagram of a particular case.
In this work we will present some techniques for ``reconstructing''
these ``missing diagrams'' in a more or less canonical way, and we
apply them to two factorizations of geometric morphisms that appear
in Part A of Johnstone's ``Sketches of an Elephant: A Topos Theory
Compendium''. Moreover, we show how to use some very visual
particular cases to develop intuition about what these factorization
``mean''.
\end{abstract}
I started using the expression ``for children'' a long time ago, at
first informally. I realized that toposes could be the right tool to
study a variation of Non-Standard Analysis in which the ultrafilters
were replaced by filters, and I tried 𝐢{very hard} to read
\cite{Johnstone} and \cite{Goldblatt}. I did not go very far, and I
kept saying to my friends ``𝐢{I need a version for children of
this!}''. My problem was that I felt that for stylistical reasons
99\% of the diagrams were omitted from text, and the examples were
mentioned very briefly or not at all... those books were intended for
``adult'' readers who knew --- maybe from contact with the ``oral
culture'' of the area? --- how to produce the ``missing'' diagrams,
examples, and calculations easily by themselves.
In this work we will see a method that can be used to produce these
``missing diagrams'' in a somewhat canonical way. When we define
``children'' precisely, not in the sense of 𝐢{who they are} but in the
sense of 𝐢{what kinds of tools and examples they prefer when they try
to learn something that is too abstract}, we get guidelines for what
kinds of concrete cases we should look for. This is my current
definition of ``children''; it turned out to be incredibly fruitful.
``Children'':
1) have trouble with very abstract definitions;
2) prefer to start from particular cases, and then generalize;
3) handle diagrams better than algebraic notation;
4) like finite objects that can be drawn explicitly;
3) Become familiar with new mathematical objects by calculating and
checking several cases, rather than by proving theorems;
4) Are not very good with algebra or proofs;
5) Are willing to use ``tools for children'' lke
It turned out that this definition of
If we establish that
``children'' have favourite 𝐢{shapes} for drawing their categorical
diagrams, then they will draw the diagrams for the general case and
for a particular case in parallel in similar shapes, in a way that
lets them 𝐢{transfer knowledge} between the general and the particular
cases quite easily; and the same between the ``external diagrams'' and
the ``internal diagrams'' of section \ref{internal-diagrams}.
In strictly mathematical terms this work is almost trivial. The result
sketched in section \ref{two-factorizations}, that certain
factorizations of geometric morphisms can be performed without leaving
the realm of ZToposes, seems to be new, and the handful of experts to
whom I showed the way of drawing sheaves in section
\ref{sheaves-on-2CGs} told me that that was easy to believe, but
they've never seen that in print and they didn't think it was
folklore.
\bsk
like to draw
their categorical diagrams all in
It turned out that
In this work we will reuse some ideas from \cite{OchsIDARCT}, that was
mostly about how to 𝐢{erase} and then 𝐢{reconstruct} information from
proofs; in particular, its sections 10 and 11 are about what happens
when an author discovers a theorem, publishes it, and then a reader
reads that, fills up the gaps in what was left implicit, and (sort of)
reconstructs in his mind the author's intuitions. Here we will take a
much more solid, or harder, view on how this reconstruction process
works.
% (find-angg ".emacs" "idarct-preprint")
% (find-idarctpage 12 "10. Transmission")
% (find-idarcttext 12 "10. Transmission")
% (find-idarctpage 13 "11. Intuition")
% (find-idarcttext 13 "11. Intuition")
The following 𝐢{definition} of ``children'' turned out to the
especially fruitful:
% (find-LATEX "catsem-u.bib" "bib-Johnstone")
% (find-LATEX "catsem-u.bib" "bib-Goldblatt" "NOT THERE")
% (find-angg "LATEX/2019ebl-abs.tex")
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\end{document}
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