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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
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\setlength{\parindent}{0em}
% _____ _ _ _
% |_ _(_) |_| | ___
% | | | | __| |/ _ \
% | | | | |_| | __/
% |_| |_|\__|_|\___|
%
% «title-page» (to ".title-page")
% (find-es "tex" "huge")
\begin{center}
{\Large {\bf Visualizing Geometric Morphisms}}
% {\large {\bf (An application of ``Logic for Children'')}}
{\large
An application of the ``Logic for Children''
project to Category Theory
}
\msk
\ColorGray{(talk @ ``Logic and Categories'' workshop, UniLog 2018)}
\bsk
%$$
\text{By:}
\quad
\begin{tabular}{c}
% (xz "~/LATEX/2018vichy-video-edrx.jpg")
\includegraphics[height=50pt]{2018vichy-video-edrx.jpg} \\
Eduardo \\ Ochs \\ (UFF, Brazil)
\end{tabular}
\quad
\begin{tabular}{c}
% (xz "~/LATEX/2018vichy-video-selana.jpg")
\includegraphics[height=50pt]{2018vichy-video-selana.jpg} \\
Selana \\ Ochs \\ \\
\end{tabular}
% \quad
% \begin{tabular}[b]{c}
% % (xz "~/LATEX/2018vichy-video-lucatelli.jpg")
% \includegraphics[width=2cm]{2018vichy-video-lucatelli.jpg} \\
% Fernando \\ Lucatelli \\
% \end{tabular}
%$$
\end{center}
\newpage
% _____ _ _ _ _
% | ___|__ _ __ ___| |__ (_) | __| |_ __ ___ _ __
% | |_ / _ \| '__| / __| '_ \| | |/ _` | '__/ _ \ '_ \
% | _| (_) | | | (__| | | | | | (_| | | | __/ | | |
% |_| \___/|_| \___|_| |_|_|_|\__,_|_| \___|_| |_|
%
% «LCT-for-children» (to ".LCT-for-children")
% (vgsp 2 "LCT-for-children")
% (vgs "LCT-for-children")
\noedrxfooter
{\bf Logic / categories / toposes for children}
({\sl Very} short version; for the long version see
the resources for the ``Logic for Children'' workshop)
\msk
\par Many years ago...
\par Non-Standard Analysis
% \par $→$ Toposes
\par $→$ Johnstone's ``Topos Theory''
\par $→$ FAR too abstract for me
\par $→$ {\bf I NEED A VERSION \ColorRed{FOR CHILDREN} OF THIS}
\msk
\ColorRed{For Children:} using ``internal views'' and
examples with finite objects that are easy to draw
Heyting Algebras that are subset of $\Z^2$ \ColorGray{(paper)}
Presheaves that can be drawn on a subset of $\Z^2$ \ColorGray{(new)}
\newpage
% ______ _ _
% |__ / | | | / \ ___
% / /| |_| | / _ \ / __|
% / /_| _ |/ ___ \\__ \
% /____|_| |_/_/ \_\___/
%
% «PHAfC» (to ".PHAfC")
{\bf Planar Heyting Algebras for Children}
($↑$ paper submitted in 2017 ---
\url{http://angg.twu.net/math-b.html})
\msk
\unitlength=8pt
\def\closeddot{\circle*{0.7}}
$
\antitabular
\begin{tabular}[c]{l}
Main definition: \\
A ZHA is a finite subset of $\Z^2$ \\
made of all even points ($x+y=2k$) \\
between $(0,0)$ and $⊤$ \\
between a ``left'' and a ``right wall''. \\
(The ``Z'' in \ColorRed{Z}HA means ``$⊂\Z^2$'') \\[5pt]
Main theorems: \\
every ZHA is a Heyting Algebra \\
every ZHA is a topology in disguise \\
\end{tabular}
\qquad
\begin{tabular}{c}
$\picturedotsa(-3,0)(3,7){
0,6
-1,5 1,5
0,4 2,4
-1,3 1,3
-2,2 0,2 2,2
-1,1 1,1
0,0
}
$ \\[32pt]
$↑$ a ZHA
\end{tabular}
$
\newpage
% ______ _ _ ____
% |__ / | | | / \ ___ |___ \
% / /| |_| | / _ \ / __| __) |
% / /_| _ |/ ___ \\__ \ / __/
% /____|_| |_/_/ \_\___/ |_____|
%
% «PHAfC-2» (to ".PHAfC-2")
{\bf Planar Heyting Algebras for Children}
($↑$ Very good paper! No prerequisites!
Lots of fun! Go read it!)
\msk
%R local PQaoi =
%R 1/ T \, 1/ T \, 1/ T \
%R | . . | | . . | | . |
%R | . . . | | . . . | | . . |
%R | . o . i | | . o . . | |d . n|
%R |. P . . .| |. P . . .| | P . |
%R | . . Q . | | . . Q . | \ F /
%R | . a . | | . . . |
%R | . . | | . . |
%R \ F / \ F /
%R local T = {a="(∧)", o="(∨)", i="(\\!→\\!)", n="(¬)", d="(\\!\\!¬¬\\!)",
%R T="·", F="·", T="⊤", F="⊥", }
%R PQaoi:tozmp({def="PQaoi", scale="12pt", meta=nil}):addcells(T):addcontour():output()
%R PQaoi:tozmp({def="lozfive", scale="12pt", meta=nil}):addlrs():addcontour():output()
\pu
$
\antitabular
\begin{tabular}[c]{l}
Most toposes have more \\
than two truth-values and \\
an intuitionistic logic. \\[5pt]
The paper PHAfC shows how \\
to visualize this (on ZHAs). \\
It uses LR-coordinates and \\
shows how the `$→$' on ZHAs \\
can be calculated quickly \\
using a formula with four cases. \\
% It doesn't mention categories. \\
\end{tabular}
% \quad
\resizebox{!}{2.2cm}{$
\begin{array}{c}
\lozfive
\qquad\qquad
\qquad\qquad
\\[-10pt]
\qquad\qquad
\qquad\qquad
\PQaoi
\end{array}
$}
$
\newpage
% ____ _ _ _ __ ____ ____
% | _ \| | | | / \ / _|/ ___| |___ \
% | |_) | |_| | / _ \ | |_| | __) |
% | __/| _ |/ ___ \| _| |___ / __/
% |_| |_| |_/_/ \_\_| \____| |_____|
%
% «local-operators» (to ".local-operators")
{\bf Planar Heyting Algebras for Children 2:}
{\bf Local Operators}
%L mp = mpnew({def="ZQuotients"}, "1R2R3212RL1"):addlrs():addcuts("c 4321/0 0123|45|6"):output()
$
\pu
\antitabular
\begin{tabular}[c]{l}
The second paper in the series. \\[5pt]
Sheaves {\sl correspond} to local \\
operators on HAs. \\[5pt]
A local operator on a ZHA \\
corresponds to slashing the ZHA \\
by diagonal cuts and blurring \\
the distinction between \\
the truth-values in each region. \\[5pt]
PHAfC doesn't mention categories. \\
PHAfC2 doesn't mention categories \ColorRed{yet}. \\
\end{tabular}
% \quad
\resizebox{!}{2.2cm}{$
\begin{array}{c}
\ZQuotients
\end{array}
$}
$
\newpage
% _________ _
% |__ / ___|__ _| |_ ___
% / / | / _` | __/ __|
% / /| |__| (_| | |_\__ \
% /____\____\__,_|\__|___/
%
% «ZCategories» (to ".ZCategories")
% (vgmp 6 "ZCategories")
% (vgm "ZCategories")
{\bf ZCategories}
%D diagram ZCatPB
%D 2Dx 100 +20 +20
%D 2D 100 o1
%D 2D
%D 2D +20 o2 o3
%D 2D
%D 2D +20 o4 o5
%D 2D
%D ren o1 o2 o3 o4 o5 ==> 1 2 3 4 5
%D
%D (( o1 o2 -> o1 o3 -> o1 o4 ->
%D o2 o3 -> o2 o4 -> o3 o5 -> o4 o5 ->
%D
%D ))
%D enddiagram
%D diagram ZPresheaf
%D 2Dx 100 +20 +20
%D 2D 100 o1
%D 2D
%D 2D +20 o2 o3
%D 2D
%D 2D +20 o4 o5
%D 2D
%D ren o1 o2 o3 o4 o5 ==> F_1 F_2 F_3 F_4 F_5
%D
%D (( o1 o2 -> o1 o3 -> o1 o4 ->
%D o2 o3 -> o2 o4 -> o3 o5 -> o4 o5 ->
%D
%D ))
%D enddiagram
\pu
$
\antitabular
\begin{tabular}[c]{l}
Choose a finite subset of $\Z^2$. \\
(Optional step: rename its points.) \\
Use this set as the set of objects \\
of a category. \\
Add a finite set of arrows. \\
This is a \ColorRed{ZCategory}. \\
The $\Z^2$-coordinates tell \\
how to draw it. \\
\end{tabular}
\quad
\diag{ZCatPB}
$
\newpage
% _________ _
% |__ / _ \ _ __ ___ ___| |__ ___ __ ___ _____ ___
% / /| |_) | '__/ _ \/ __| '_ \ / _ \/ _` \ \ / / _ \/ __|
% / /_| __/| | | __/\__ \ | | | __/ (_| |\ V / __/\__ \
% /____|_| |_| \___||___/_| |_|\___|\__,_| \_/ \___||___/
%
% «ZPresheaves» (to ".ZPresheaves")
% (vgmp 7 "ZPresheaves")
% (vgm "ZPresheaves")
{\bf ZPresheaves and ZToposes}
A ZPresheaf is a functor $F:\catA → \Set$,
where $\catA$ is a ZCategory.
(Obs: not $F:\catA^{\ColorRed\op} → \Set$!)
A ZPresheaf $F$ inherits its drawing instructions from $\catA$.
\ColorGray{(``Positional notations'')}
%
$$
\resizebox{!}{35pt}{$
\catA =
\left(
\diag{ZCatPB}
\right)
\quad
F =
\left(
\diag{ZPresheaf}
\right)
$}
$$
A ZTopos is a category $\Set^\catA$ where $\catA$ is a ZCategory.
\newpage
% ___ _ _
% |_ _|_ __ | |_ ___ _ __ _ __ __ _| |
% | || '_ \| __/ _ \ '__| '_ \ / _` | |
% | || | | | || __/ | | | | | (_| | |
% |___|_| |_|\__\___|_| |_| |_|\__,_|_|
%
% «internal-views» (to ".internal-views")
% (find-LATEX "2018vichy-video.tex" "internal-views")
% (find-LATEX "2018vichy-video.tex" "internal-views-2")
{\bf Internal views}
(Part 1: functions)
\unitlength=10pt
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 a0 a5 midpoint .TeX= \myoval(3.4,10)(1.7,5)[1.5] place
%D b-1 b5 midpoint .TeX= \myoval(3.4,11)(1.7,5.5)[1.5] 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
$$\scalebox{0.7}{$\diag{second-blob-function}$}
$$
(`$↦$'s take elements of a blob-set to another blob-set)
\newpage
% ___ _ _ ____
% |_ _|_ __ | |_ ___ _ __ _ __ __ _| | |___ \
% | || '_ \| __/ _ \ '__| '_ \ / _` | | __) |
% | || | | | || __/ | | | | | (_| | | / __/
% |___|_| |_|\__\___|_| |_| |_|\__,_|_| |_____|
%
% «internal-views-2» (to ".internal-views-2")
% (vgmp 9 "internal-views-2")
% (vgm "internal-views-2")
{\bf Internal views}
(Part 2: functors)
Internal views of \ColorRed{functors} have blob-\ColorRed{categories}
instead of blob-\ColorRed{sets}. Compare:
%D diagram internal-view-F
%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
%D diagram internal-view-F-noblobs
%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
$$\scalebox{0.7}{$\diag{second-blob-function}$}
\qquad
\qquad
\scalebox{0.7}{$\diag{internal-view-F}$}
$$
\newpage
% ___ _ _ _____
% |_ _|_ __ | |_ ___ _ __ _ __ __ _| | |___ /
% | || '_ \| __/ _ \ '__| '_ \ / _` | | |_ \
% | || | | | || __/ | | | | | (_| | | ___) |
% |___|_| |_|\__\___|_| |_| |_|\__,_|_| |____/
%
% «internal-views-3» (to ".internal-views-2")
{\bf Internal views}
(Part 3: omitting the blobs)
$$\scalebox{0.9}{$\diag{internal-view-F}$}
\qquad
\quad
\scalebox{0.9}{$\diag{internal-view-F-noblobs}$}
$$
\newpage
% ___ _ _ _ _
% |_ _|_ __ | |_ ___ _ __ _ __ __ _| | | || |
% | || '_ \| __/ _ \ '__| '_ \ / _` | | | || |_
% | || | | | || __/ | | | | | (_| | | |__ _|
% |___|_| |_|\__\___|_| |_| |_|\__,_|_| |_|
%
% «internal-views-4» (to ".internal-views-4")
% (vgsp 11 "internal-views-4")
% (vgs "internal-views-4")
{\bf Internal views}
(Part 4: adjunctions)
Left: generic adjunction $L⊣R$
Middle: generic geometric morphism $f^*⊣f_*$
Right: g.m. between toposes $\Set^\catA$ and $\Set^\catB$
%
%D diagram adjunctions
%D 2Dx 100 +25 +30 +25 +30 +30
%D 2D 100 A0 A1 B0 B1 C0 C1
%D 2D
%D 2D +20 A2 A3 B2 B3 C2 C3
%D 2D
%D 2D +15 A4 A5 B4 B5 C4 C5
%D 2D
%D 2D +20 C6 C7
%D 2D
%D ren A0 A1 A2 A3 A4 A5 ==> LC C D RD \catD \catC
%D ren B0 B1 B2 B3 B4 B5 ==> f^*F F G f_*G \mathcal{E} \mathcal{F}
%D ren C0 C1 C2 C3 C4 C5 ==> f^*F F G f_*G \Set^\catA \Set^\catB
%D ren C6 C7 ==> \catA \catB
%D
%D (( A0 A1 <-|
%D A0 A2 -> A1 A3 ->
%D A2 A3 |->
%D A4 A5 <- sl^ .plabel= a L
%D A4 A5 -> sl_ .plabel= b R
%D A0 A3 harrownodes nil 20 nil <->
%D
%D B0 B1 <-|
%D B0 B2 -> B1 B3 ->
%D B2 B3 |->
%D B4 B5 <- sl^ .plabel= a f^*
%D B4 B5 -> sl_ .plabel= b f_*
%D B0 B3 harrownodes nil 20 nil <->
%D
%D C0 C1 <-|
%D C0 C2 -> C1 C3 ->
%D C2 C3 |->
%D C4 C5 <- sl^ .plabel= a f^*
%D C4 C5 -> sl_ .plabel= b f_*
%D C0 C3 harrownodes nil 20 nil <->
%D C6 C7 -> .plabel= a f
%D
%D
%D ))
%D enddiagram
%D
$$\pu
\diag{adjunctions}
$$
\newpage
% ____ _ _ _
% | _ \ __ _ _ __ __ _| | | ___| |
% | |_) / _` | '__/ _` | | |/ _ \ |
% | __/ (_| | | | (_| | | | __/ |
% |_| \__,_|_| \__,_|_|_|\___|_|
%
% «parallel» (to ".parallel")
% (vgmp 12 "parallel")
% (vgm "parallel")
% (vivp 10 "children")
% (viv "children")
{\bf Working in two languages in parallel}
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
{\bf Working in two universes in parallel}
In Non-Standard Analysis we have {\sl transfer theorems}
$$\ptm{c}{Standard \\ universe}
\two/<-`->/<500>
\ptm{c}{Non-Standard \\ universe \\ (ultrapower)}
$$
\newpage
% _ _ ____ __ __
% / |___| |_ / ___| \/ |
% | / __| __| | | _| |\/| |
% | \__ \ |_ | |_| | | | |
% |_|___/\__| \____|_| |_|
%
% «first-gm» (to ".first-gm")
% (vgsp 14 "first-gm")
% (vgs "first-gm")
{\bf Our first geometric morphism}
%
%L sesw = {[" w"]="↙", [" e"]="↘"}
%
%R local B, F, RG = 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, G, LF = 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 B :tozmp({def="pB", scale="7pt", meta="s p"}):addcells(sesw):output()
%R F :tozmp({def="pF", scale="7pt", meta="s p"}):addcells(sesw):output()
%R RG:tozmp({def="pRG", scale="7pt", meta="s p"}):addcells(sesw):output()
%R A :tozmp({def="pA", scale="7pt", meta="s p"}):addcells(sesw):output()
%R G :tozmp({def="pG", scale="7pt", meta="s p"}):addcells(sesw):output()
%R LF:tozmp({def="pLF", scale="7pt", meta="s p"}):addcells(sesw):output()
\def\Gt{G_2 {×_{G_4}} G_3}
\pu
%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 2D +25 A6 A7
%D 2D
%D ren A0 A1 ==> \pLF \pF
%D ren A2 A3 ==> \pG \pRG
%D ren A4 A5 ==> \Set^\catA \Set^\catB
%D ren A6 A7 ==> \pA \pB
%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 A6 A7 -> sl^ .plabel= a 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
$$\pu
\resizebox{!}{70pt}{$
\begin{array}{ccc}
\diag{GM-children-big}&
\qquad
\qquad&
\diag{GM-general}\\
\\
\text{(for children; inclusion, sheaf)}&&
\text{(for adults)}\\
\end{array}
$}
$$
\newpage
% _ __ _ _ _ _
% / \ / _| __ _ ___| |_ ___ _ __(_)______ _| |_(_) ___ _ __
% / _ \ | |_ / _` |/ __| __/ _ \| '__| |_ / _` | __| |/ _ \| '_ \
% / ___ \ | _| (_| | (__| || (_) | | | |/ / (_| | |_| | (_) | | | |
% /_/ \_\ |_| \__,_|\___|\__\___/|_| |_/___\__,_|\__|_|\___/|_| |_|
%
% «a-factorization» (to ".a-factorization")
% (vgmp 15 "a-factorization")
% (vgm "a-factorization")
{\bf A factorization}
Elephant $=$ Bible
Section A4: Geometric Morphisms
Each `$\diagxyto/->/<200>$' below is a g.m. (an adjunction)
Any g.m. factors as a surjection followed by an inclusion.
Any inclusion factors as a dense g.m.
followed by a closed g.m.\,.
%
%D diagram ??
%D 2Dx 100 +30 +30 +30
%D 2D 100 A0 A3
%D 2D
%D 2D +12 B0 B1 B3
%D 2D
%D 2D +12 C1 C2 C3
%D 2D
%D ren A0 A3 ==> \calA \calD
%D ren B0 B1 B3 ==> \calA \calB \calD
%D ren C1 C2 C3 ==> \calB \calC \calD
%D
%D (( A0 A3 -> .plabel= a \text{any}
%D B0 B1 -> .plabel= a \text{surjection}
%D B1 B3 -> .plabel= a \text{inclusion}
%D C1 C2 -> .plabel= a \text{dense}
%D C2 C3 -> .plabel= a \text{close}
%D
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
The Elephant {\sl constructs} the toposes $\calB$, $\calC$ and the maps.
\newpage
% _____ _ _ _ _ ____
% | ___|_ _ ___| |_ ___ _ __(_)______ _| |_(_) ___ _ __ |___ \
% | |_ / _` |/ __| __/ _ \| '__| |_ / _` | __| |/ _ \| '_ \ __) |
% | _| (_| | (__| || (_) | | | |/ / (_| | |_| | (_) | | | | / __/
% |_| \__,_|\___|\__\___/|_| |_/___\__,_|\__|_|\___/|_| |_| |_____|
%
% «factorization-2» (to ".factorization-2")
% (vgmp 16 "factorization-2")
% (vgm "factorization-2")
{\bf A factorization: version using ZPresheaves}
This would be a nicer theorem --- that if we start
with ZToposes $\Set^\catA$ and $\Set^\catD$ the factorization can be
through ZToposes...
\def\ftext#1#2{\text{$#1$ (#2)}}
\def\ftext#1#2{\text{#2}}
%L forths["=="] = function () pusharrow("==") end
%
%D diagram ??
%D 2Dx 100 +35 +35 +35
%D 2D 100 A0 A3
%D 2D
%D 2D +20 B0 B1 B3
%D 2D
%D 2D +20 C1 C2 C3
%D 2D
%D 2D +20 D2
%D 2D
%D ren A0 A3 ==> \Set^\catA \Set^\catD
%D ren B0 B1 B3 ==> \Set^\catA \calB \Set^\catD
%D ren C1 C2 C3 ==> \Set^\catB \calC \Set^\catD
%D ren D2 ==> \Set^\catC
%D
%D (( A0 A3 -> .plabel= a \ftext{a}{any}
%D B0 B1 -> .plabel= a \ftext{s}{surjection}
%D B1 B3 -> .plabel= a \ftext{i}{inclusion}
%D C1 C2 -> .plabel= a \ftext{d}{dense}
%D C2 C3 -> .plabel= a \ftext{c}{closed}
%D A0 B0 = A3 B3 = B1 C1 == B3 C3 = C2 D2 ==
%D ))
%D enddiagram
%D
$$\pu
\resizebox{!}{50pt}{$
\diag{??}
$}
$$
\newpage
% _____ _ _ _ _ _____
% | ___|_ _ ___| |_ ___ _ __(_)______ _| |_(_) ___ _ __ |___ /
% | |_ / _` |/ __| __/ _ \| '__| |_ / _` | __| |/ _ \| '_ \ |_ \
% | _| (_| | (__| || (_) | | | |/ / (_| | |_| | (_) | | | | ___) |
% |_| \__,_|\___|\__\___/|_| |_/___\__,_|\__|_|\___/|_| |_| |____/
%
% «factorization-3» (to ".factorization-3")
% (visp 15)
% _____ _ _ _ _ _ _
% | ___|_ _ ___| |_ ___ _ __(_)______ _| |_(_) ___ _ __ | || |
% | |_ / _` |/ __| __/ _ \| '__| |_ / _` | __| |/ _ \| '_ \ | || |_
% | _| (_| | (__| || (_) | | | |/ / (_| | |_| | (_) | | | | |__ _|
% |_| \__,_|\___|\__\___/|_| |_/___\__,_|\__|_|\___/|_| |_| |_|
%
% «factorization-4» (to ".factorization-4")
% (visp 16)
{\bf That factorization, for children}
We start with a particular case, with a factorization
that only has ZToposes, and we use it to understand
how the Elephant defines sujection, inclusion, etc...
($s$ is not an inclusion, $i$ is not a surjection, and so on)
%
\def\ftext#1#2{\text{#2}}
\def\ftext#1#2{\text{$#1$ (#2)}}
%
%D diagram fact-children
%D 2Dx 100 +55 +45 +45
%D 2D 100 F G H I
%D 2D
%D 2D +15 A0 A3
%D 2D
%D 2D +20 B0 B1 B3
%D 2D
%D 2D +20 C1 C2 C3
%D 2D
%D ren A0 A3 ==> \Set^\catA \Set^\catD
%D ren B0 B1 B3 ==> \Set^\catA \Set^\catB \Set^\catD
%D ren C1 C2 C3 ==> \Set^\catB \Set^\catC \Set^\catD
%D
%D (( F place G place H place I place
%D A0 A3 -> .plabel= a \ftext{g}{any}
%D B0 B1 -> .plabel= a \ftext{s}{surjection}
%D B1 B3 -> .plabel= a \ftext{i}{inclusion}
%D C1 C2 -> .plabel= a \ftext{d}{dense}
%D C2 C3 -> .plabel= a \ftext{c}{closed}
%D A0 B0 = A3 B3 = B1 C1 = B3 C3 =
%D ))
%D enddiagram
%
\pu
$$
\resizebox{!}{50pt}{$
\diag{fact-children}
$}
$$
\newpage
% ____ _ __ _ _
% / ___| _ _ _ __(_) / / (_)_ __ ___| |
% \___ \| | | | '__| | / / | | '_ \ / __| |
% ___) | |_| | | | | / / | | | | | (__| |
% |____/ \__,_|_| _/ | /_/ |_|_| |_|\___|_|
% |__/
%
% «surjection-inclusion» (to ".surjection-inclusion")
% (ele "elephant-A4.2.6")
% (vgmp 18 "surjection-inclusion")
% (vgm "surjection-inclusion")
{\bf The surjection-inclusion factorization for children}
%
%D diagram dense-closed
%D 2Dx 100 +55 +20 +40 +20 +120
%D 2D 100 AD0 <----------------| AD1
%D 2D
%D 2D +25 AD2 |----------------> AD3
%D 2D
%D 2D +15 AD4 -----------------> AD5
%D 2D
%D 2D +20 AB0 AB1 ABa BDa BD0 BD1
%D 2D
%D 2D +20 AB2 AB3 ABb BDb BD2 BD3
%D 2D
%D 2D +15 AB4 AB5 BD4 BD5
%D 2D
%D ren AD0 AD1 ==> g^*I I
%D ren AD2 AD3 ==> F g_*F
%D ren AD4 AD5 ==> \Set^\catA \Set^\catD
%D
%D ren AB0 AB1 ==> s^*G G
%D ren AB2 AB3 ==> F s_*F
%D ren AB4 AB5 ==> \Set^\catA \Set^\catB
%D ren ABa ABb ==> G s_*s^*G
%D
%D ren BD0 BD1 ==> i^*I I
%D ren BD2 BD3 ==> G i_*G
%D ren BD4 BD5 ==> \Set^\catB \Set^\catD
%D ren BDa BDb ==> i^*i_*G G
%D
%D (( AD0 AD1 <-|
%D AD0 AD2 ->
%D AD1 AD3 ->
%D AD2 AD3 |->
%D AD0 AD3 harrownodes nil 20 nil <->
%D AD4 AD5 -> .plabel= a \ftext{g}{any}
%D
%D AB0 AB1 <-|
%D ABa ABb -> .plabel= r \sm{ηG\\\text{(monic)}}
%D AB0 AB2 ->
%D AB1 AB3 ->
%D AB2 AB3 |->
%D AB0 AB3 harrownodes nil 20 nil <->
%D AB4 AB5 -> .plabel= a \ftext{s}{surjection}
%D
%D BDa BDb -> .plabel= l \sm{εG\\\text{(iso)}}
%D BD0 BD1 <-|
%D BD0 BD2 ->
%D BD1 BD3 ->
%D BD2 BD3 |->
%D BD0 BD3 harrownodes nil 20 nil <->
%D BD4 BD5 -> .plabel= a \ftext{i}{inclusion}
%D
%D ))
%D enddiagram
\pu
$$
% \resizebox{!}{60pt}{$
\resizebox{220pt}{!}{$
\diag{dense-closed}
$}
$$
\newpage
% ____ __ _ _
% | _ \ ___ _ __ ___ ___ / / ___| | ___ ___ ___ __| |
% | | | |/ _ \ '_ \/ __|/ _ \ / / / __| |/ _ \/ __|/ _ \/ _` |
% | |_| | __/ | | \__ \ __/ / / | (__| | (_) \__ \ __/ (_| |
% |____/ \___|_| |_|___/\___| /_/ \___|_|\___/|___/\___|\__,_|
%
% «dense-closed» (to ".dense-closed")
% (vgmp 19 "dense-closed")
% (vgm "dense-closed")
% (visp 16)
% (vis )
{\bf The dense-closed factorization for children}
%
%D diagram dense-closed
%D 2Dx 100 +20 +45 +20 +40 +45
%D 2D 100 BDa BD0 <----------------| BD1
%D 2D
%D 2D +25 BDb BD2 |----------------> BD3
%D 2D
%D 2D +15 BD4 -----------------> BD5
%D 2D
%D 2D +20 BC0 BC1 BCa CD0 CD1
%D 2D
%D 2D +20 BC2 BC3 BCb CD2 CD3
%D 2D
%D 2D +20 BC4 BC5 CD4 CD5
%D 2D
%D 2D +20
%D 2D
%D ren BDa BDb ==> i^*i_*G G
%D ren BD0 BD1 ==> i^*I I
%D ren BD2 BD3 ==> G i_*G
%D ren BD4 BD5 ==> \Set^\catB \Set^\catD
%D
%D ren BCa BCb ==> K d_*d^*K
%D ren BC0 BC1 ==> d^*H H
%D ren BC2 BC3 ==> G d_*G
%D ren BC4 BC5 ==> \Set^\catB \Set^\catC
%D
%D ren CD0 CD1 ==> c^*I I
%D ren CD2 CD3 ==> H c_*H
%D ren CD4 CD5 ==> \Set^\catC \Set^\catD
%D
%D (( BDa BDb -> .plabel= l \sm{εG\\\text{(iso)}}
%D BD0 BD1 <-|
%D BD0 BD2 ->
%D BD1 BD3 ->
%D BD2 BD3 |->
%D BD0 BD3 harrownodes nil 20 nil <->
%D BD4 BD5 -> .plabel= a \ftext{i}{inclusion}
%D
%D BCa BCb -> .plabel= r \sm{ηK\\\text{(monic)}}
%D BC0 BC1 <-|
%D BC0 BC2 ->
%D BC1 BC3 ->
%D BC2 BC3 |->
%D BC0 BC3 harrownodes nil 20 nil <->
%D BC4 BC5 -> .plabel= a \ftext{d}{dense}
%D
%D CD0 CD1 <-|
%D CD0 CD2 ->
%D CD1 CD3 ->
%D CD2 CD3 |->
%D CD0 CD3 harrownodes nil 20 nil <->
%D CD4 CD5 -> .plabel= a \ftext{c}{closed}
%D
%D ))
%D enddiagram
%
\pu
$$
\resizebox{!}{60pt}{$
\diag{dense-closed}
$}
$$
\ColorGray{($K$ is a constant ZPresheaf in $\Set^\catC$)}
\newpage
{\bf Acoording to the Elephant...}
%D diagram mysterious-B-and-C
%D 2Dx 100 +45 +45 +40
%D 2D 100 A0 A3
%D 2D
%D 2D +20 B0 B1 B3
%D 2D
%D 2D +20 c1
%D 2D
%D 2D +20 C1 C2 C3
%D 2D
%D 2D +20 d2
%D 2D
%D 2D +20 D2
%D 2D
%D ren A0 A3 ==> \Set^\catA \Set^\catD
%D ren B0 B1 B3 ==> \Set^\catA \calB \Set^\catD
%D ren C1 C2 C3 ==> \Set^\catB \calC \Set^\catD
%D ren D2 ==> \Set^\catC
%D
%D ren c1 ==> (\Set^\catB)_\bbG
%D ren d2 ==> \mathsf{sh}_{¬¬}(\Set^\catD)\oque
%D
%D (( A0 A3 -> .plabel= a \ftext{a}{any}
%D B0 B1 -> .plabel= a \ftext{s}{surjection}
%D B1 B3 -> .plabel= a \ftext{i}{inclusion}
%D C1 C2 -> .plabel= a \ftext{d}{dense}
%D C2 C3 -> .plabel= a \ftext{c}{closed}
%D A0 B0 = A3 B3 = B1 c1 == c1 C1 == B3 C3 = C2 d2 == d2 D2 ==
%D ))
%D enddiagram
%D
\pu
\def\oque{\;\;\;(???)}
\antitabular
\begin{tabular}{l}
A4.2.7, 4.2.10: \\
to build $\calB$ we need \\
comonads and \\
coalgebras \\
\\
A4.5.9, A4.5.20: \\
$\calC = \mathsf{sh}_{¬¬}(\Set^\catD)$ \\
\ColorRed{(can't be!)} \\
\end{tabular}
\quad
$\resizebox{!}{60pt}{$
\diag{mysterious-B-and-C}
$}
$
\newpage
{\bf Another strategy}
Start with a functor $g:\catA → \catD$.
It induces a geometric morphism $g^*⊣g_*$.
$g^*$ is trivial to build.
$g_*$ can be found by guess-and-test.
\ColorGray{(or by Kan extensions)}
\msk
The functor $g$ can:
collapse objects, $(1 \;\;\; 2) → (1)$
create objects, $(\,) → (3)$
collapse arrows, $(4 \two/->`->/ 5) → (4 → 5)$
create arrows, $(6 \;\;\; 7) → (6 → 7)$
\msk
Try to factor it. Example: if $g$ just collapses objects...
\newpage
{\bf Another strategy}
The functor $g$ can do several \ColorRed{things}:
collapse objects, $(1 \;\;\; 2) → (1)$
create objects, $(\,) → (3)$
collapse arrows, $(4 \two/->`->/ 5) → (4 → 5)$
create arrows, $(6 \;\;\; 7) → (6 → 7)$
refine the order, $(2 → 4) → (1 → 2 → 3 → 4 → 5)$...
\msk
Try to factor it.
Example: if $g$ just \ColorRed{collapses} objects,
then it factor as $s=g$ (surj. part), $i=\id$ (inclusion part)...
The factorization filters the {\sl things} that the functor can do,
\ColorRed{collapsing objects} go to the surjective part.
\newpage
{\bf Another strategy}
\antitabular
\begin{tabular}{l}
Choose a functor \\
$f:\catA→\catB$ \\
that does all \ColorRed{things}. \\
Factorize it. \\
$\calB$ and $\calC$ will \\
be non-trivial. \\
They tell us how \\
$\calB$ and $\calC$ will be \\
\ColorRed{modulo} \\
\ColorRed{isomorphism}. \\
\end{tabular}
\;\;
$\resizebox{!}{60pt}{$
\diag{mysterious-B-and-C}
$}
$
\newpage
For more information:
\url{http://angg.twu.net/logic-for-children-2018.html}
\url{http://angg.twu.net/math-b.html}
% (ph2 "algebra-of-J-ops")
% (ph2p 32 "algebra-of-J-ops")
% (ph2p 34 "algebra-of-J-ops" "algebra")
\end{document}
% Local Variables:
% coding: utf-8-unix
% ee-tla: "vgs"
% End: