% (find-angg "LATEX/2017vichy-slides-1.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2017vichy-slides-1.tex"))
% (defun d () (interactive) (find-xpdfpage "~/LATEX/2017vichy-slides-1.pdf"))
% (defun e () (interactive) (find-LATEX "2017vichy-slides-1.tex"))
% (defun u () (interactive) (find-latex-upload-links "2017vichy-slides-1"))
% (find-xpdfpage "~/LATEX/2017vichy-slides-1.pdf")
% (find-sh0 "cp -v  ~/LATEX/2017vichy-slides-1.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2017vichy-slides-1.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2017vichy-slides-1.pdf
%               file:///tmp/2017vichy-slides-1.pdf
%           file:///tmp/pen/2017vichy-slides-1.pdf
% http://angg.twu.net/LATEX/2017vichy-slides-1.pdf
\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")
%
\begin{document}

\catcode`\^^J=10
\directlua{dednat6dir = "dednat6/"}
\directlua{dofile(dednat6dir.."dednat6.lua")}
\directlua{texfile(tex.jobname)}
\directlua{verbose()}
\directlua{output(preamble1)}
\def\expr#1{\directlua{output(tostring(#1))}}
\def\eval#1{\directlua{#1}}
\def\pu{\directlua{pu()}}

\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

\def\frown{=(}
\def\M#1#2{#1↦#2}

\def\hc#1#2 #3 {((#1,#2),\{#3\})}
\def\hC#1#2 #3 {((#1,#2),\csm{#3})}


{\bf A non-category}

Let $\catC$ be category with $\Objs_{\catC} = \{1,2,3,4\}$,

and these non-identity arrows:

%D diagram ??
%D 2Dx     100   +30   +30
%D 2D  100 1 --> 2
%D 2D
%D 2D  +30       3 --> 4
%D 2D
%D (( 1 2 -> .plabel= a 12
%D    2 3 -> .plabel= m 23
%D    3 4 -> .plabel= b 34
%D    1 3 -> .plabel= l 123
%D    2 4 -> .plabel= a 234
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$

The arrows from 1 to 4 are not shown --- $\Hom_\catC(1,4) = \{12304, 10234\}$.

The identity arrows are $\id_\catC(1) = 11$, $\id_\catC(2) = 22$,
$\id_\catC(3) = 33$, $\id_\catC(4) = 44$.

The composition is trivial --- for example, $12;23 = 123$, and $11;12=12$

everywhere except here:

$(12;23);34 = 123;34 = 12304$ and

$12;(23;34) = 12;234 = 10234$ \qquad $⇐$ associativity fails!

\bsk

We have
$$
  \Hom_\catC = \cmat{
    \hc11 11 , & \hc12 12 , & \hc13 123 , & \hC14 {12304,\\10234} , \\
    \hc21 \, , & \hc22 22 , & \hc23 23  , & \hc24 234 , \\
    \hc31 \, , & \hc32 \, , & \hc33 33  , & \hc34  34 , \\
    \hc41 \, , & \hc42 \, , & \hc43 \,  , & \hc44  44   \\
  }
$$
%
but $({;_\catC})$ is too big to show...

We can look at parts of if, though:

$({;_\catC})_{1,2,3}(12,23) = 123$,

$({;_\catC})_{1,2,3} = \{((12,23),123)\}$,

$((1,2,3), ((12,23),123))∈({;_\catC})$.







\newpage




{\bf Solving factorization problems}

%D diagram ??
%D 2Dx     100 +40 +30 +40 +30 +40
%D 2D  100 A1  A2  B1  B2  C1  C2
%D 2D
%D 2D  +30     A3      B3      C3
%D 2D
%D 2D  +20         D1  D2  E1  E2
%D 2D
%D 2D  +30             D3      E3
%D 2D
%D 2D  +20         F1  F2  G1  G2
%D 2D
%D 2D  +30             F3      G3
%D 2D
%D ren  A1 A2 A3   ==>   \{1,2,3\} \{4,5,6\} \{7,8,9\}
%D ren  B1 B2 B3   ==>   \{1,2,3\} \{4,5,6\} \{7,8,9\}
%D ren  C1 C2 C3   ==>   \{1,2,3\} \{4,5,6\} \{7,8,9\}
%D ren  D1 D2 D3   ==>   \{1\}     \{2,3\}   \{4\}
%D ren  E1 E2 E3   ==>   \{1,2\}   \{3,4\}   \{5,6\}
%D ren  F1 F2 F3   ==>   \{1,2\}   \{3,4\}   \{5,6\}
%D ren  G1 G2 G3   ==>   \{1,2\}   \{3,4\}   \{5,6\}
%D (( A1 A2 -> .plabel= a \sm{\M14\\\M26\\\M35}
%D    A2 A3 -> .plabel= r \sm{\M47\\\M57\\\M69}
%D    A1 A3 -> .plabel= l ?
%D
%D    B1 B2 -> .plabel= a ?
%D    B2 B3 -> .plabel= r \sm{\M47\\\M58\\\M69}
%D    B1 B3 -> .plabel= l \sm{\M17\\\M28\\\M39}
%D
%D    C1 C2 -> .plabel= a \sm{\M14\\\M25\\\M36}
%D    C2 C3 -> .plabel= r ?
%D    C1 C3 -> .plabel= l \sm{\M17\\\M28\\\M39}
%D
%D    D1 D2 -> sl^ .plabel= a ?
%D    D1 D2 -> sl_ .plabel= b ?
%D    D2 D3 -> .plabel= r \sm{\M24\\\M34}
%D    D1 D3 -> .plabel= l \sm{\M14}
%D
%D    E1 E2 -> .plabel= a \sm{\M13\\\M23}
%D    E2 E3 -> sl_ .plabel= l ?
%D    E2 E3 -> sl^ .plabel= r ?
%D    E1 E3 -> .plabel= l \sm{\M15\\\M25}
%D
%D    F1 F2 -> .plabel= a \frown
%D    F2 F3 -> .plabel= r \sm{\M35\\\M45}
%D    F1 F3 -> .plabel= l \sm{\M15\\\M26}
%D
%D    G1 G2 -> .plabel= a \sm{\M13\\\M23}
%D    G2 G3 -> .plabel= r \frown
%D    G1 G3 -> .plabel= l \sm{\M15\\\M26}
%D    
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$



`$\diagxyto/->/^{?}$': exactly one solution

`$\two/->`->/^{?}_{?}$': two different solutions

`$\diagxyto/->/^{\frown}$': no solutions


\newpage

{\bf Solving factorization problems: products}

{\bf Exercise.} Complete:

%D diagram ??
%D 2Dx     100 +35 +35 +30 +30 +30 +30
%D 2D  100     A1          E1
%D 2D
%D 2D  +30 A2  A3  A4  E2  E3  E4
%D 2D
%D 2D  +30     B1          F1
%D 2D
%D 2D  +30 B2  B3  B4  F2  F3  F4
%D 2D
%D 2D  +30     C1          G1
%D 2D
%D 2D  +30 C2  C3  C4  G2  G3  G4
%D 2D
%D 2D  +30     D1          H1
%D 2D
%D 2D  +30 D2  D3  D4  H2  H3  H4
%D 2D
%D ren  A1 A2 A3 A4  ==>  \{1,2\} \{3,4\} \csm{35,53,\\45,46} \{5,6\}
%D ren  B1 B2 B3 B4  ==>  \{1,2\} \{3,4\} \csm{35,53,\\45,46} \{5,6\}
%D ren  C1 C2 C3 C4  ==>  \{1,2\} \{3,4\} \csm{35,53,\\45,46} \{5,6\}
%D ren  D1 D2 D3 D4  ==>  \{1,2\} \{3,4\} \csm{35,53,\\45,46} \{5,6\}
%D ren  E1 E2 E3 E4  ==>  \{1,2\} \{3,4\} \csm{(3,5),(3,6),\\(4,5),(4,6)} \{5,6\}
%D (( A1 A2 -> .plabel= l ?
%D    A1 A3 -> .plabel= m \sm{\Mm135\\\Mm253}
%D    A1 A4 -> .plabel= r ?
%D    A2 A3 <- .plabel= b \sm{\MM353\\\MM533\\\MM454\\\MM464}
%D    A3 A4 -> .plabel= b \sm{\MM355\\\MM535\\\MM455\\\MM466}
%D
%D    B1 B2 -> .plabel= l \sm{\M14\\\M24}
%D    B1 B3 -> .plabel= m ?
%D    B1 B4 -> .plabel= r \sm{\M15\\\M26}
%D    B2 B3 <- .plabel= b \sm{\MM353\\\MM533\\\MM454\\\MM464}
%D    B3 B4 -> .plabel= b \sm{\MM355\\\MM535\\\MM455\\\MM466}
%D
%D    C1 C2 -> .plabel= l \sm{\M13\\\M24}
%D    C1 C3 -> sl_ .plabel= l ?
%D    C1 C3 -> sl^ .plabel= r ?
%D    C1 C4 -> .plabel= r \sm{\M15\\\M26}
%D    C2 C3 <- .plabel= b \sm{\MM353\\\MM533\\\MM454\\\MM464}
%D    C3 C4 -> .plabel= b \sm{\MM355\\\MM535\\\MM455\\\MM466}
%D
%D    D1 D2 -> .plabel= l \sm{\M14\\\M24}
%D    D1 D3 -> .plabel= m \frown
%D    D1 D4 -> .plabel= r \sm{\M15\\\M26}
%D    D2 D3 <- .plabel= b \sm{\MM353\\\MM533\\\MM454\\\MM464}
%D    D3 D4 -> .plabel= b \sm{\MM355\\\MM535\\\MM455\\\MM466}
%D    
%D ))
%D enddiagram
%D
$$\pu
  \def\Mm#1#2#3{#1↦#2#3}
  \def\MM#1#2#3{#1#2↦#3}
  \def\MP#1#2#3{#1↦#3}
  \diag{??}
$$


% `$\diagxyto/->/^{?}$': exactly one solution
% 
% `$\two/->`->/^{?}_{?}$': two different solutions
% 
% `$\diagxyto/->/^{\frown}$': no solutions


\newpage

{\bf A question of style}






\end{document}

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