% (find-angg "LATEX/2018vichy-video.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2018vichy-video.tex"))
% (defun d () (interactive) (find-xpdfpage "~/LATEX/2018vichy-video.pdf"))
% (defun b () (interactive) (find-zsh "bibtex 2018vichy-video; makeindex 2018vichy-video"))
% (defun e () (interactive) (find-LATEX "2018vichy-video.tex"))
% (defun u () (interactive) (find-latex-upload-links "2018vichy-video"))
% (find-xpdfpage "~/LATEX/2018vichy-video.pdf")
% (find-sh0 "cp -v  ~/LATEX/2018vichy-video.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2018vichy-video.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2018vichy-video.pdf
%               file:///tmp/2018vichy-video.pdf
%           file:///tmp/pen/2018vichy-video.pdf
% http://angg.twu.net/LATEX/2018vichy-video.pdf
%
% (find-es "ffmpeg" "vichy-video")

% «.colors»		(to "colors")
% «.myoval»		(to "myoval")
% «.title-page»		(to "title-page")
% «.why»		(to "why")
% «.why-2»		(to "why-2")
% «.why-3»		(to "why-3")
% «.bigger-project»	(to "bigger-project")
% «.adults»		(to "adults")
% «.children»		(to "children")
% «.children-2»		(to "children-2")
% «.toolbox»		(to "toolbox")
% «.publish»		(to "publish")
% «.publish-2»		(to "publish-2")
% «.rest-adults»	(to "rest-adults")
% «.VGM»		(to "VGM")
% «.VGM-2»		(to "VGM-2")
% «.VGM-3»		(to "VGM-3")
% «.VGM-4»		(to "VGM-4")
% «.VGM-5»		(to "VGM-5")
% «.VGM-6»		(to "VGM-6")
% «.internal-views»	(to "internal-views")
% «.internal-views-functors»	(to "internal-views-functors")
% «.internal-views-functors-2»	(to "internal-views-functors-2")
% «.internal-views-gm»		(to "internal-views-gm")
% «.internal-views-gm-2»	(to "internal-views-gm-2")

\documentclass[oneside]{book}
\usepackage[colorlinks]{hyperref} % (find-es "tex" "hyperref")
%\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
\usepackage{color}                % (find-LATEX "edrx15.sty" "colors")
\usepackage{colorweb}             % (find-es "tex" "colorweb")
%\usepackage{tikz}
%
% (find-dn6 "preamble6.lua" "preamble0")
%\usepackage{proof}   % For derivation trees ("%:" lines)
\input diagxy        % For 2D diagrams ("%D" lines)
%\xyoption{curve}     % For the ".curve=" feature in 2D diagrams
%
\usepackage{edrx15}               % (find-angg "LATEX/edrx15.sty")
\input edrxaccents.tex            % (find-angg "LATEX/edrxaccents.tex")
\input edrxchars.tex              % (find-LATEX "edrxchars.tex")
\input edrxheadfoot.tex           % (find-dn4ex "edrxheadfoot.tex")
\input edrxgac2.tex               % (find-LATEX "edrxgac2.tex")
%
% (find-angg ".emacs.papers" "latexgeom")
% (find-LATEXfile "2016-2-GA-VR.tex" "{geometry}")
% (find-latexgeomtext "total={6.5in,8.75in},")
\usepackage[paperwidth=11cm, paperheight=8.5cm,
            %total={6.5in,4in},
            %textwidth=4in,  paperwidth=4.5in,
            %textheight=5in, paperheight=4.5in,
            %a4paper,
            top=1.5cm, bottom=.5cm, left=1cm, right=1cm, includefoot
           ]{geometry}
%
\begin{document}

\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"}  % (find-LATEX "dednat6load.lua")

%\directlua{dofile "edrxtikz.lua"} % (find-LATEX "edrxtikz.lua")
%\directlua{dofile "edrxpict.lua"} % (find-LATEX "edrxpict.lua")
%L V.__tostring = function (v) return format("(%.3f,%.3f)", v[1], v[2]) end

% «colors» (to ".colors")
% (find-LATEX "2017ebl-slides.tex" "colors")
% (find-LATEX "2017ebl-slides.tex" "colors" "\\def\\ColorGreen")
\long\def\ColorRed   #1{{\color{Red}#1}}
\long\def\ColorViolet#1{{\color{MagentaVioletLight}#1}}
\long\def\ColorGreen #1{{\color{SpringDarkHard}#1}}
\long\def\ColorGreen #1{{\color{SpringGreenDark}#1}}
\long\def\ColorGray  #1{{\color{GrayLight}#1}}

% «myoval» (to ".myoval")
% (find-LATEXfile "2018pict2e.tex" "\\def\\myoval")
\def\myvcenter#1{\ensuremath{\vcenter{\hbox{#1}}}}%
\def\myoval(#1,#2)(#3,#4)[#5]{%
  \myvcenter{%
    \begin{picture}(#1,#2)(-#3,-#4)
      \put(0,0){\oval[#5](#1,#2)}
    \end{picture}%
  }}

\def\calU{{\mathcal{U}}}
\def\calI{{\mathcal{I}}}

\setlength{\parindent}{0em}


%  _____ _ _   _      
% |_   _(_) |_| | ___ 
%   | | | | __| |/ _ \
%   | | | | |_| |  __/
%   |_| |_|\__|_|\___|
%                     
% «title-page» (to ".title-page")
% (vivp 1 "title-page")
% (viv    "title-page")

{\Huge {\bf Logic for Children}}

(i.e., for people without mathematical maturity ---

a workshop at UniLog 2018)
%
$$
\begin{tabular}[b]{c}
  % (xz              "~/LATEX/2018vichy-video-edrx.jpg")
  \includegraphics[width=2cm]{2018vichy-video-edrx.jpg} \\
  Eduardo \\ Ochs \\
\end{tabular}
\quad
\begin{tabular}[b]{c}
  % (xz              "~/LATEX/2018vichy-video-lucatelli.jpg")
  \includegraphics[width=2cm]{2018vichy-video-lucatelli.jpg} \\
  Fernando \\ Lucatelli \\
\end{tabular}
\quad
\begin{tabular}[b]{c}
  % (xz              "~/LATEX/2018vichy-video-selana.jpg")
  \includegraphics[width=2cm]{2018vichy-video-selana.jpg} \\
  Selana \\ Ochs \\
\end{tabular}
$$


\newpage

\noedrxfooter

% __        ___          ___ 
% \ \      / / |__  _   |__ \
%  \ \ /\ / /| '_ \| | | |/ /
%   \ V  V / | | | | |_| |_| 
%    \_/\_/  |_| |_|\__, (_) 
%                   |___/    
%
% «why» (to ".why")
% (vivp 2)

{\bf Why?}

\ssk

\par Many years ago...
\par Non-Standard Analysis
\par $→$ Ultrapowers
\par $→$ Filter-powers
\par $→$ Toposes
\par $→$ Johnstone's ``Topos Theory''

\newpage

% __        ___          ___   ____  
% \ \      / / |__  _   |__ \ |___ \ 
%  \ \ /\ / /| '_ \| | | |/ /   __) |
%   \ V  V / | | | | |_| |_|   / __/ 
%    \_/\_/  |_| |_|\__, (_)  |_____|
%                   |___/            
%
% «why-2» (to ".why-2")
% (vivp 3)

{\bf Why?}

\ssk

\par Many years ago...
\par Non-Standard Analysis
\par $→$ Ultrapowers
\par $→$ Filter-powers
\par $→$ Toposes
\par $→$ Johnstone's ``Topos Theory''
\par $→$ FAR too abstract for me

\newpage

% __        ___          ___   _____ 
% \ \      / / |__  _   |__ \ |___ / 
%  \ \ /\ / /| '_ \| | | |/ /   |_ \ 
%   \ V  V / | | | | |_| |_|   ___) |
%    \_/\_/  |_| |_|\__, (_)  |____/ 
%                   |___/            
%
% «why-3» (to ".why-3")
% (vivp 4)

{\bf Why?}

\ssk

\par Many years ago...
\par Non-Standard Analysis
\par $→$ Ultrapowers
\par $→$ Filter-powers
\par $→$ Toposes
\par $→$ Johnstone's ``Topos Theory''
\par $→$ FAR too abstract for me
\par $→$ {\bf I NEED A VERSION FOR CHILDREN OF THIS}

\newpage

% __        ___          ___   _____ 
% \ \      / / |__  _   |__ \ |___ / 
%  \ \ /\ / /| '_ \| | | |/ /   |_ \ 
%   \ V  V / | | | | |_| |_|   ___) |
%    \_/\_/  |_| |_|\__, (_)  |____/ 
%                   |___/            
%
% «why-3» (to ".why-3")
% (vivp 5)

{\bf Why?}

\ssk

\par Many years ago...
\par Non-Standard Analysis
\par $→$ Ultrapowers
\par $→$ Filter-powers
\par $→$ Toposes
\par $→$ Johnstone's ``Topos Theory''
\par $→$ FAR too abstract for me
\par $→$ {\bf I NEED A VERSION \ColorRed{FOR CHILDREN} OF THIS}

\newpage

%  ____            _           _   
% |  _ \ _ __ ___ (_) ___  ___| |_ 
% | |_) | '__/ _ \| |/ _ \/ __| __|
% |  __/| | | (_) | |  __/ (__| |_ 
% |_|   |_|  \___// |\___|\___|\__|
%               |__/               
%
% «bigger-project» (to ".bigger-project")
% (vivp 6)

With time this became

\ssk

{\Large {\bf A MUCH BIGGER}}

project...

\msk

Some subtasks:

{\def\Sub#1{}

1. Find the right definition of ``children''

\Sub{(inspired by how I function)}

2. Develop a basic toolbox

\Sub{(and name its tools)}

3. Make these things publishable

\Sub{(make them look formal and non-trivial)}

}

\newpage

% (vivp 7 "bigger-project")
% (viv    "bigger-project")

With time this became

\ssk

{\Large {\bf A MUCH BIGGER}}

project...

\msk

Some subtasks:

{\def\Sub#1{\quad\ColorGray{#1}}

1. Find the right definition of ``children''

\Sub{(inspired by how I function)}

2. Develop a basic toolbox

\Sub{(and name its tools)}

3. Make these things publishable

\Sub{(make them look formal and non-trivial)}

}

\newpage

%     _       _       _ _       
%    / \   __| |_   _| | |_ ___ 
%   / _ \ / _` | | | | | __/ __|
%  / ___ \ (_| | |_| | | |_\__ \
% /_/   \_\__,_|\__,_|_|\__|___/
%                               
% «adults» (to ".adults")
% (vivp 8)

{\bf The opposite of ``children''}

The opposite of ``children'' is

``\ColorRed{adults}'', or ``\ColorRed{mathematicians}''.

% An ``adult'' feels that everything

A ``mathematician'' feels that everything

should be done as generally and as abstractly

as possible --- and doing otherwise is {\sl bad style}.

\newpage

% (vivp 9)

{\bf The opposite of ``children''}

The opposite of ``children'' is

``\ColorRed{adults}'', or ``\ColorRed{mathematicians}''.

% An ``adult'' feels that everything

A ``mathematician'' feels that everything

should be done as generally and as abstractly

as possible --- and doing otherwise is {\sl bad style}.

\msk

Example: finding a right adjoint by

guesswork / trial and error...

\msk

One expression that I love is: ``{\sl this step

(or argument) offends adults}''.





\newpage

%   ____ _     _ _     _                
%  / ___| |__ (_) | __| |_ __ ___ _ __  
% | |   | '_ \| | |/ _` | '__/ _ \ '_ \ 
% | |___| | | | | | (_| | | |  __/ | | |
%  \____|_| |_|_|_|\__,_|_|  \___|_| |_|
%                                       
% «children» (to ".children")
% (vivp 10 "children")
% (viv     "children")

{\bf Task 1: The right definition of ``children'':}

They prefer to start from particular cases

and then generalize ---

They like diagrams and finite objects

drawn very explicitly ---

They become familiar with mathematical ideas

by calculating / checking several cases

(rather than by proving theorems)

\newpage

% «children-2»  (to ".children-2")
% (vivp 11 "children-2")
% (viv     "children-2")

{\bf Task 1: The right definition of ``children'':}

They prefer to start from particular cases

and then generalize ---

They like diagrams and finite objects

drawn very explicitly ---

They become familiar with mathematical ideas

by calculating / checking several cases

(rather than by proving theorems)

\msk

% http://puzzler.sourceforge.net/docs/pentominoes.html
% http://puzzler.sourceforge.net/docs/images/ominoes/pentominoes-8x8.png

$\hskip-5.5pt
 %
 \begin{tabular}[b]{l}
   Example: pentominos. \\
   Let ``children'' \ColorRed{play} \\
   with pentominos for a while \\
   \ColorRed{before} showing to them \\
   theorems and game trees! \\
  \end{tabular}
  %
  \qquad
  \quad
  %
  \includegraphics[height=52pt]{pentominoes-8x8.png}
$

\newpage

%  _____           _ _               
% |_   _|__   ___ | | |__   _____  __
%   | |/ _ \ / _ \| | '_ \ / _ \ \/ /
%   | | (_) | (_) | | |_) | (_) >  < 
%   |_|\___/ \___/|_|_.__/ \___/_/\_\
%                                    
% «toolbox» (to ".toolbox")
% (vivp 12)

{\bf Task 2: Develop a basic toolbox}

I'm starting with ``Category Theory for children''

because I am a categorist, and

because CT uses diagrams and generalizations {\sl a lot}...

\msk

Basic tools:

Use \ColorRed{parallel diagrams},

\ColorRed{positional notations},

\ColorRed{internal views},

\ColorGray{archetypal cases}...

\msk

\ColorGreen{(I'll show some diagrams soon)}

\newpage

%  ____        _     _ _     _     
% |  _ \ _   _| |__ | (_)___| |__  
% | |_) | | | | '_ \| | / __| '_ \ 
% |  __/| |_| | |_) | | \__ \ | | |
% |_|    \__,_|_.__/|_|_|___/_| |_|
%                                  
% «publish» (to ".publish")
% (vivp 13)

{\bf Task 3: Find ways to publish this}

CT books treat examples very briefly,

as if they were trivial exercises... ${=}($

Ideas: do things ``for children'' and ``for adults''

in parallel, find ways to {\sl transfer knowledge}

between the two approaches...

\msk

\ColorGreen{(Non-standard Analysis has transfer theorems

between the standard universe, $\Set$, and $\Set^\calU/\calI$)}

\newpage

%  ____        _     _ _     _       ____  
% |  _ \ _   _| |__ | (_)___| |__   |___ \ 
% | |_) | | | | '_ \| | / __| '_ \    __) |
% |  __/| |_| | |_) | | \__ \ | | |  / __/ 
% |_|    \__,_|_.__/|_|_|___/_| |_| |_____|
%                                          
% «publish-2» (to ".publish-2")
% (vivp 14)

{\bf Task 3:  Find ways to publish this}

CT books treat examples very briefly,

as if they were trivial exercises... ${=}($

Ideas: do things ``for children'' and ``for adults''

in \ColorRed{parallel}, find ways to \ColorRed{\sl transfer knowledge}

between the two approaches...
%
\def\tm #1#2{       \begin{tabular}{#1}#2\end{tabular}}
\def\ptm#1#2{\left (\begin{tabular}{#1}#2\end{tabular}\right )}
\def\smm#1#2{\sm{\text{#1}\\\text{#2}}}
%
$$\ptm{c}{particular \\ case \\ ``for children''}
  \two/<-`->/<500>^{\smm{particularize}{(easy)}}_{\smm{generalize}{(hard)}}
  \ptm{c}{general \\ case \\ ``for adults''}
$$

The diagrams for the general case and for a particular case

{\sl have the same shape!!!}

\newpage

%                _        __            _       _ _       
%  _ __ ___  ___| |_     / /   __ _  __| |_   _| | |_ ___ 
% | '__/ _ \/ __| __|   / /   / _` |/ _` | | | | | __/ __|
% | | |  __/\__ \ |_   / /   | (_| | (_| | |_| | | |_\__ \
% |_|  \___||___/\__| /_/     \__,_|\__,_|\__,_|_|\__|___/
%                                                         
% «rest-adults» (to ".rest-adults")
% (vivp 15)

{\bf In the rest of these slides...}

...we will show an example:

\ColorRed{Geometric Morphisms} for children!

($↑$ a thing from Topos Theory)



\newpage

% __     ______ __  __ 
% \ \   / / ___|  \/  |
%  \ \ / / |  _| |\/| |
%   \ V /| |_| | |  | |
%    \_/  \____|_|  |_|
%                      
% «VGM» (to ".VGM")
% (vivp 16)

{\bf Visualizing Geometric Morphisms}

An application:

\ColorRed{Sheaves} and \ColorRed{Geometric Morphisms}

$↑$ two parts of Topos Theory that look

{\sl incredibly abstract} at first

% \msk

\ColorGray{
(Btw, I'll give a talk at the ``Logic and Categories''

workshop about that)
}

\msk

Trick:

Start with presheaves {\sl that are easy to visualize;}

Start with a very small, planar category like this...

\newpage


% __     ______ __  __   ____  
% \ \   / / ___|  \/  | |___ \ 
%  \ \ / / |  _| |\/| |   __) |
%   \ V /| |_| | |  | |  / __/ 
%    \_/  \____|_|  |_| |_____|
%                              
% «VGM-2» (to ".VGM-2")
% (vivp 17)
% (find-angg "LUA/texinfo.lua" "preproc")

{\bf Visualizing Geometric Morphisms}

\ColorRed{Trick: positional notations}

Start with presheaves {\sl that are easy to visualize;}

Start with a very small, planar category like this,
%
%L sesw = {[" w"]="↙",  [" e"]="↘"}
%
%R local B, BF = 3/       1             \, 3/      F_1            \
%R                |    w     e          |   |    w     e          |
%R                | 2           3       |   |F_2         F_3      |
%R                |    e     w     e    |   |    e     w     e    |
%R                |       4           5 |   |      F_4         F_5|
%R                |          e     w    |   |          e     w    |
%R                \             6       /   \            F_6      /
%R
%R B:tozmp({def="Bbig", scale="10pt", meta="p"}):addcells(sesw):output()
$$\pu \catB = \Bbig
$$

{\color{GrayLight}
Technicalities:

$\catB$ is a preorder
}

\newpage

% __     ______ __  __   _____ 
% \ \   / / ___|  \/  | |___ / 
%  \ \ / / |  _| |\/| |   |_ \ 
%   \ V /| |_| | |  | |  ___) |
%    \_/  \____|_|  |_| |____/ 
%                              
% «VGM-3» (to ".VGM-3")
% (vivp 18)

{\bf Visualizing Geometric Morphisms}

...and now a presheaf $F$ on $\catB$

can be drawn like this...
%
%R local B, BF = 3/       1             \, 3/      F_1            \
%R                |    w     e          |   |    w     e          |
%R                | 2           3       |   |F_2         F_3      |
%R                |    e     w     e    |   |    e     w     e    |
%R                |       4           5 |   |      F_4         F_5|
%R                |          e     w    |   |          e     w    |
%R                \             6       /   \            F_6      /
%R
%R B :tozmp({def="Bmed", scale="7pt", meta="s p"}):addcells(sesw):output()
%R BF:tozmp({def="BF",   scale="7pt", meta="s p"}):addcells(sesw):output()
$$\pu \catB = \Bmed \qquad F = \BF
$$

\newpage

% __     ______ __  __   _  _   
% \ \   / / ___|  \/  | | || |  
%  \ \ / / |  _| |\/| | | || |_ 
%   \ V /| |_| | |  | | |__   _|
%    \_/  \____|_|  |_|    |_|  
%                               
% «VGM-4»  (to ".VGM-4")
% (vivp 19)

{\bf Visualizing Geometric Morphisms}

...and now a presheaf $F$ on $\catB$

can be drawn like this...
%
%R local B, BF = 3/       1             \, 3/      F_1            \
%R                |    w     e          |   |    w     e          |
%R                | 2           3       |   |F_2         F_3      |
%R                |    e     w     e    |   |    e     w     e    |
%R                |       4           5 |   |      F_4         F_5|
%R                |          e     w    |   |          e     w    |
%R                \             6       /   \            F_6      /
%R
%R B :tozmp({def="Bmed", scale="7pt", meta="s p"}):addcells(sesw):output()
%R BF:tozmp({def="BF",   scale="7pt", meta="s p"}):addcells(sesw):output()
$$\pu \catB = \Bmed \qquad F = \BF
$$

{\color{GrayLight}
Technicalities:

$F_1, F_2, \ldots, F_6$ are sets,

the `$F_i→F_j$'s are functions,

$F:\catB→\Set$, i.e., $F∈\Set^{\catB}$,

And there may be an `$\op$' omitted somewhere
}


% \newpage
% 
% {\bf Visualizing Geometric Morphisms}
% 
% ...and now a presheaf $F$ on $\catB$
% 
% can be drawn like this...
% %
% %R local B, BF = 3/       1             \, 3/      F_1            \
% %R                |    w     e          |   |    w     e          |
% %R                | 2           3       |   |F_2         F_3      |
% %R                |    e     w     e    |   |    e     w     e    |
% %R                |       4           5 |   |      F_4         F_5|
% %R                |          e     w    |   |          e     w    |
% %R                \             6       /   \            F_6      /
% %R
% %R B :tozmp({def="Bmed", scale="7pt", meta="s p"}):addcells(sesw):output()
% %R BF:tozmp({def="BF",   scale="7pt", meta="s p"}):addcells(sesw):output()
% $$\pu \catB = \Bmed \qquad F = \BF
% $$
 
\newpage

% __     ______ __  __   ____  
% \ \   / / ___|  \/  | | ___| 
%  \ \ / / |  _| |\/| | |___ \ 
%   \ V /| |_| | |  | |  ___) |
%    \_/  \____|_|  |_| |____/ 
%                              
% «VGM-5»  (to ".VGM-5")
% (vivp 20)

{\bf Visualizing Geometric Morphisms}

...choose a subcategory $\catA$ of $\catB$, e.g., the one below.

Then a presheaf $G$ on $\catA$ can be drawn as:
%
%R local B, BF, BG = 3/       1             \, 3/      F_1            \, 3/      !Gt            \
%R                    |    w     e          |   |    w     e          |   |    w     e          |
%R                    | 2           3       |   |F_2         F_3      |   |G_2         G_3      |
%R                    |    e     w     e    |   |    e     w     e    |   |    e     w     e    |
%R                    |       4           5 |   |      F_4         F_5|   |      G_4         G_5|
%R                    |          e     w    |   |          e     w    |   |          e     w    |
%R                    \             6       /   \            F_6      /   \             1       /
%R
%R local A, AG, AF = 3/ 2           3       \, 3/G_2         G_3      \, 3/F_2         F_3      \
%R                    |    e     w     e    |   |    e     w     e    |   |    e     w     e    |
%R                    \       4           5 /   \      G_4         G_5/   \      F_4         F_5/
%R                
%R
%R B :tozmp({def="Bmed", scale="7pt", meta="s p"}):addcells(sesw):output()
%R BF:tozmp({def="BF",   scale="7pt", meta="s p"}):addcells(sesw):output()
%R BG:tozmp({def="BG",   scale="7pt", meta="s p"}):addcells(sesw):output()
%R A :tozmp({def="Amed", scale="7pt", meta="s p"}):addcells(sesw):output()
%R AG:tozmp({def="AG",   scale="7pt", meta="s p"}):addcells(sesw):output()
%R AF:tozmp({def="AF",   scale="7pt", meta="s p"}):addcells(sesw):output()
\def\Gt{G_2 {×_{G_4}} G_3}
\pu
$$\catB = \Bmed \qquad F = \BF
$$
$$\catA = \Amed \qquad G = \AG
$$


{\color{GrayLight}
Technicalities: too many ${=}($
}

\newpage
% __     ______ __  __    __   
% \ \   / / ___|  \/  |  / /_  
%  \ \ / / |  _| |\/| | | '_ \ 
%   \ V /| |_| | |  | | | (_) |
%    \_/  \____|_|  |_|  \___/ 
%                              
% «VGM-6»  (to ".VGM-6")
% (vivp 21)

{\bf Visualizing Geometric Morphisms}

...and the inclusion $f:\catA→\catB$

induces a geometric morphism $f:\Set^\catA→\Set^\catB$,

that ``is'' an adjunction $f^*⊣f_*$:
%
$$\Set^\catA
  \two/<-`->/<200>^{f^*}_{f_*}
  \Set^\catB
$$

...where $f^*$ is ``obvious'' \ColorGray{(for some value of ``obvious'')}

and $f_*$ can be obtained by \ColorRed{trial and error} if we don't

understand Kan Extensions...

Kan Extensions: \ColorRed{for adults}

Trial and error: \ColorRed{for children}

\newpage

%  ___       _                        _ 
% |_ _|_ __ | |_ ___ _ __ _ __   __ _| |
%  | || '_ \| __/ _ \ '__| '_ \ / _` | |
%  | || | | | ||  __/ |  | | | | (_| | |
% |___|_| |_|\__\___|_|  |_| |_|\__,_|_|
%                                       
% «internal-views» (to ".internal-views")
% (vivp 22 "internal-views")
% (viv     "internal-views")

{\bf Interlude: internal views}

The best way to explain the adjunction of the

previous slide to children is through \ColorRed{\sl internal views}.

The internal view of the \ColorRed{function} $√{}:\N→\R$ is:
%
\def\ooo(#1,#2){\begin{picture}(0,0)\put(0,0){\oval(#1,#2)}\end{picture}}
\def\oooo(#1,#2){{\setlength{\unitlength}{1ex}\ooo(#1,#2)}}
%
%D diagram second-blob-function
%D 2Dx     100 +20   +20   
%D 2D  100 a-1 |-->  b-1    
%D 2D  +08 a0  |-->  b0    
%D 2D  +08 a1  |-->  b1    
%D 2D  +08 a2  |-->  b2    
%D 2D  +08 a3  |-->  b3    
%D 2D  +08 a4  |-->  b4    
%D 2D  +14 a5  |-->  b5    
%D 2D  +25 \N  --->  \R
%D 2D
%D ren a-1 a0 a1 a2 a3 a4 a5 ==> -1 0 1 2 3 4 n
%D ren b-1 b0 b1 b2 b3 b4 b5 ==> -1 0 1 \sqrt{2} \sqrt{3} 2 \sqrt{n}
%D ((  a0 a5 midpoint .TeX= \oooo(7,23) y+= -2 place
%D    b-1 b5 midpoint .TeX= \oooo(7,25) y+= -2 place
%D       b-1 place
%D    a0 b0 |->
%D    a1 b1 |->
%D    a2 b2 |->
%D    a3 b3 |->
%D    a4 b4 |->
%D    a5 b5 |->
%D    \N \R -> .plabel= a \sqrt{\phantom{a}}
%D    a-1 relplace -7 -7 \phantom{foo}
%D    b5  relplace  7  7 \phantom{bar}
%D ))
%D enddiagram
%D
$$\pu
  \resizebox{2.2cm}{!}{$\diag{second-blob-function}$}
$$

(`$↦$'s take elements of a blob-set to another blob-set)

% Internal views of \ColorRed{functors} have blob-{\sl categories}.


\newpage
% «internal-views-functors»  (to ".internal-views-functors")

% (vivp 23)

{\bf Interlude: internal views}

Internal views of \ColorRed{functors} have blob-\ColorRed{categories}

instead of blob-\ColorRed{sets}, like this:


\unitlength=10pt

%D diagram ??
%D 2Dx     100    +40
%D 2D  100 A      FA
%D 2D
%D 2D  +30 B      FB
%D 2D
%D 2D  +20 \catC  \catD
%D 2D
%D
%D (( A FA |->
%D    B FB |->
%D    A FB harrownodes nil 18 nil |->
%D    A B -> .plabel= l g
%D    FA FB -> .plabel= r Fg
%D    \catC \catD -> .plabel= a F
%D    A  B  midpoint .TeX= \myoval(3.4,7)(1.7,3.5)[1] place
%D    FA FB midpoint .TeX= \myoval(3.4,7)(1.7,3.5)[1] place
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$

\newpage
% «internal-views-functors-2»  (to ".internal-views-functors-2")

% (vivp 24)

{\bf Interlude: internal views}

We draw the internal view of $F:\catC → \catD$ as this,

%D diagram ??
%D 2Dx     100    +25
%D 2D  100 A      FA
%D 2D
%D 2D  +20 B      FB
%D 2D
%D 2D  +15 \catC  \catD
%D 2D
%D
%D (( A FA |->
%D    B FB |->
%D    # A FB harrownodes nil 18 nil |->
%D    A B -> .plabel= l g
%D    FA FB -> .plabel= r Fg
%D    \catC \catD -> .plabel= a F
%D    # A  B  midpoint .TeX= \myoval(3.4,7)(1.7,3.5)[1] place
%D    # FA FB midpoint .TeX= \myoval(3.4,7)(1.7,3.5)[1] place
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$

\msk

we omit the blobs (the ``{\unitlength=5pt\myoval(1,2)(0.5,1)[0.3]}''s), and we draw

the internal view --- objects and maps in $\catC$ and $\catD$ ---

above the external view ($F:\catC→\catD$).


\newpage
% «internal-views-gm»  (to ".internal-views-gm")
% (vivp 25 "internal-views-gm")
% (viv     "internal-views-gm")

{\bf Internal views}

Here is the internal view of the

geometric morphism $f:\Set^\catA→\Set^\catB$...

remember that $f$ is an adjunction $f^*⊣f_*$.


%D diagram GM-particular
%D 2Dx     100   +35
%D 2D  100 A0    A1
%D 2D
%D 2D  +25 A2    A3
%D 2D
%D 2D  +15 A4    A5
%D 2D
%D ren A0 A1 ==> f^*F F
%D ren A2 A3 ==> G f_*G
%D ren A4 A5 ==> \Set^\catA \Set^\catB
%D
%D (( A0 A1 <-
%D    A2 A3 ->
%D    A0 A2 ->
%D    A1 A3 ->
%D    A0 A3 harrownodes nil 20 nil <->
%D    A4 A5 <- sl^ .plabel= a f^*
%D    A4 A5 -> sl_ .plabel= b f_*
%D
%D ))
%D enddiagram
%D
%D diagram GM-general
%D 2Dx     100   +35
%D 2D  100 A0    A1
%D 2D
%D 2D  +25 A2    A3
%D 2D
%D 2D  +15 A4    A5
%D 2D
%D ren A0 A1 ==> f^*F F
%D ren A2 A3 ==> G f_*G
%D ren A4 A5 ==> \calF \calE
%D
%D (( A0 A1 <-
%D    A2 A3 ->
%D    A0 A2 ->
%D    A1 A3 ->
%D    A0 A3 harrownodes nil 20 nil <->
%D    A4 A5 <- sl^ .plabel= a f^*
%D    A4 A5 -> sl_ .plabel= b f_*
%D
%D ))
%D enddiagram
%D
\pu
$$%\pu
  \resizebox{!}{50pt}{$
    \begin{array}{ccc}
      \diag{GM-particular}&
      \quad&
      \ColorGray{
      \diag{GM-general}
      }
      \\
      \\
      \ColorGray{\text{(particular case)}}&&
      \ColorGray{\text{(general case)}}\\
    \end{array}
  $}
$$


\newpage
% «internal-views-gm-2»  (to ".internal-views-gm-2")
% (vivp 26 "internal-views-gm-2")
% (viv     "internal-views-gm-2")

{\bf A geometric morphism (for children)}

%D diagram GM-children-big
%D 2Dx     100   +55
%D 2D  100 A0    A1
%D 2D
%D 2D  +45 A2    A3
%D 2D
%D 2D  +25 A4    A5
%D 2D
%D ren A0 A1 ==> \AF \BF
%D ren A2 A3 ==> \AG \BG
%D ren A4 A5 ==> \Set^\catA \Set^\catB
%D
%D (( A0 A1 <-|
%D    A2 A3 |->
%D    A0 A2 ->
%D    A1 A3 ->
%D    A0 A3 harrownodes nil 20 nil <->
%D    A4 A5 <- sl^ .plabel= a f^*
%D    A4 A5 -> sl_ .plabel= b f_*
%D
%D ))
%D enddiagram
%D
$$\pu
  \resizebox{!}{70pt}{$
     \begin{array}{ccc}
       \diag{GM-children-big}&
       \qquad
       \qquad&
       \diag{GM-general}\\
       \\
       \text{(for children)}&&
       \text{(for adults)}\\
     \end{array}
  $}
$$

\newpage
% (vivp 27)




{\bf Resources about the workshop}

Here:

\url{http://angg.twu.net/logic-for-children-2018.html}

Cheers! $=)$











\end{document}

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