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% This file: (find-LATEX "2019J-ops-classifier.tex")
% See: (find-LATEX "2020J-ops-new.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2019J-ops-classifier.tex" :end))
% (defun d () (interactive) (find-pdf-page "~/LATEX/2019J-ops-classifier.pdf"))
% (defun e () (interactive) (find-LATEX "2019J-ops-classifier.tex"))
% (defun u () (interactive) (find-latex-upload-links "2019J-ops-classifier"))
% (find-pdf-page "~/LATEX/2019J-ops-classifier.pdf")
% (find-sh0 "cp -v ~/LATEX/2019J-ops-classifier.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2019J-ops-classifier.pdf /tmp/pen/")
% file:///home/edrx/LATEX/2019J-ops-classifier.pdf
% file:///tmp/2019J-ops-classifier.pdf
% file:///tmp/pen/2019J-ops-classifier.pdf
% http://angg.twu.net/LATEX/2019J-ops-classifier.pdf
% (find-LATEX "2019.mk")
\directlua{tf_push("2019J-ops-classifier.tex")}
% «.classifier» (to "classifier")
% «.Omega-and-j» (to "Omega-and-j")
% «.pullbacks-formally» (to "pullbacks-formally")
% «.NTs-B-1» (to "NTs-B-1")
% «.NTs-B-C» (to "NTs-B-C")
% «.NTs-1-Om» (to "NTs-1-Om")
% «.NTs-C-Om» (to "NTs-C-Om")
% «.NTs-Om-Om» (to "NTs-Om-Om")
% «.fig:five-sqconds» (to "fig:five-sqconds")
% «.pullbacks-visually» (to "pullbacks-visually")
% ____ _ _ __ _
% / ___| | __ _ ___ ___(_)/ _(_) ___ _ __
% | | | |/ _` / __/ __| | |_| |/ _ \ '__|
% | |___| | (_| \__ \__ \ | _| | __/ |
% \____|_|\__,_|___/___/_|_| |_|\___|_|
%
% «classifier» (to ".classifier")
% (jonp 31 "classifier")
% (joo "classifier")
\subsection{The classifier}
Take a map $t:1→C$ in a topos. Choose a map $g:B→C$ and form the
pullback with $t$, obtaining maps $f:A→B$ and $h:A→1$. We can prove
that any map from the terminal is monic, and this implies that $t$ is
monic, and so, by a property of pullbacks, $f$ is a monic too; and $h$
is the unique map from $A$ to the terminal. In a diagram:
%
%D diagram cla-0
%D 2Dx 100 +25 +40 +25 +40 +25
%D 2D 100 A1 B0 B1 C0 C1
%D 2D
%D 2D +25 A2 A3 B2 B3 C2 C3
%D 2D
%D ren A1 A2 A3 ==> 1 B C
%D ren B0 B1 B2 B3 ==> A 1 B C
%D ren C0 C1 C2 C3 ==> A 1 B C
%D
%D (( A1 A3 -> .plabel= r t
%D A2 A3 -> .plabel= b g
%D
%D # A1 B2 midpoint .TeX= ⇒ place
%D A1 B2 harrownodes nil 20 nil =>
%D
%D B0 B1 -> .plabel= a h
%D B0 B2 -> .plabel= l f
%D B1 B3 -> .plabel= r t
%D B2 B3 -> .plabel= b g
%D B0 relplace 7 7 \pbsymbol{7}
%D
%D # B1 C2 midpoint .TeX= ⇒ place
%D B1 C2 harrownodes nil 20 nil =>
%D
%D C0 C1 -> .plabel= a !
%D C0 C2 >-> .plabel= l f
%D C1 C3 >-> .plabel= r t
%D C2 C3 -> .plabel= b g
%D C0 relplace 7 7 \pbsymbol{7}
%D ))
%D enddiagram
%D
$$\pu
\diag{cla-0}
$$
We can consider that the operation ``form the pullback with
$t:1 \monicto C$'' receives a map $g: B→C$ and returns a monic $f:
A \monicto B$ ``completing the pullback''.
Every topos has a classifier object $Ω$ and a ``true'' map
$⊤:1 \monicto Ω$ with the property that for every monic $f: A \monicto
B$ there is a unique map $χ:B→Ω$ ``completing the pullback''. In a
diagram:
%
%D diagram cla-1
%D 2Dx 100 +25 +40 +25 +45 +25 +40 +25
%D 2D 100 A0 A1 B0 B1 C0 C1 D0 D1
%D 2D
%D 2D +25 A2 A3 B2 B3 C2 C3 D2 D3
%D 2D
%D ren A0 A1 A2 A3 ==> A 1 B Ω
%D ren B0 B1 B2 B3 ==> A 1 B Ω
%D ren C0 C1 C2 C3 ==> A 1 B Ω
%D ren D0 D1 D2 D3 ==> A' 1 B Ω
%D
%D (( A0 A2 >-> .plabel= l f
%D A1 A3 >-> .plabel= r ⊤
%D
%D A1 B2 harrownodes nil 20 nil =>
%D
%D B0 B1 -> .plabel= a !
%D B0 B2 >-> .plabel= l f
%D B1 B3 >-> .plabel= r ⊤
%D B2 B3 -> .plabel= b χ
%D B0 relplace 7 7 \pbsymbol{7}
%D
%D
%D C1 C3 -> .plabel= r ⊤
%D C2 C3 -> .plabel= b χ
%D
%D C1 D2 harrownodes nil 20 nil =>
%D
%D D0 D1 -> .plabel= a !
%D D0 D2 >-> .plabel= l f'
%D D1 D3 >-> .plabel= r ⊤
%D D2 D3 -> .plabel= b χ
%D D0 relplace 7 7 \pbsymbol{7}
%D ))
%D enddiagram
%D
$$\pu
\diag{cla-1}
$$
These two operations, $f↦χ$ and $χ↦f'$, are not exactly inverse to one
another: if we apply them in the order $f↦χ↦f'$ we may obtain an $f'$
that is isomorphic to $f$ in the sense that there is an iso $A ↔ A'$
such that the triangle below commutes:
%
%D diagram cla-triangle
%D 2Dx 100 +15 +15
%D 2D 100 A A'
%D 2D
%D 2D +25 O
%D 2D
%D ren O ==> Ω
%D
%D (( A O >-> .plabel= l f
%D A' O >-> .plabel= r f'
%D A A' <->
%D ))
%D enddiagram
%D
$$\pu
\diag{cla-triangle}
$$
This is explained in \cite[p.139]{LambekScott}
% (find-lambekscottpage (+ 8 139) "subobject clasifier")
% ___ _ _
% / _ \ _ __ ___ __ _ _ __ __| | (_)
% | | | | '_ ` _ \ / _` | '_ \ / _` | | |
% | |_| | | | | | | | (_| | | | | (_| | | |
% \___/|_| |_| |_| \__,_|_| |_|\__,_| _/ |
% |__/
%
% «Omega-and-j» (to ".Omega-and-j")
% (jonp 31 "Omega-and-j")
% (joe "Omega-and-j")
\subsection{The classifier and the local operator}
We know that every category $\Set^{(P,A)}$ is a topos, but how do we
calculate and visualize its classifier object $Ω$ and the map $⊤:1→Ω$?
And what is the local operator $j:Ω→Ω$ ``associated to'' our
J-operator $J:\Sub(1)→\Sub(1)$?
\msk
%D diagram Omega-and-j
%D 2Dx 100 +30 +30
%D 2D 100 B --> 1
%D 2D | |
%D 2D v v
%D 2D +30 C --> Om1 --> Om2
%D 2D
%D ren Om1 Om2 ==> Ω Ω
%D
%D (( B 1 -> .plabel= a !
%D B C >-> .plabel= l i
%D 1 Om1 >-> .plabel= r ⊤
%D C Om1 -> .plabel= a χ_B
%D Om1 Om2 -> .plabel= a j
%D B relplace 7 7 \pbsymbol{7}
%D ))
%D enddiagram
%D
%D diagram Omega-and-j-2
%D 2Dx 100 +30 +30
%D 2D 100 B ----------> 1
%D 2D | |
%D 2D v v
%D 2D +30 C --> Om1 --> Om2
%D 2D
%D ren Om1 Om2 B ==> Ω Ω \ovl{B}
%D
%D (( B 1 -> .plabel= a !
%D B C >-> .plabel= l \ovl{i}
%D 1 Om2 >-> .plabel= r ⊤
%D C Om1 -> .plabel= a χ_B
%D Om1 Om2 -> .plabel= a j
%D B relplace 7 7 \pbsymbol{7}
%D ))
%D enddiagram
%D
\pu
% TODO: Explain the prequisites for this section. Explain that I
% learned this from Bell but McLarty is more readable.
%
% (find-books "__cats/__cats.el" "mclarty")
% (find-books "__cats/__cats.el" "bell")
We need to start by understanding two pullbacks. Remember that:
\begin{itemize}
\item $⊤:1→Ω$ has a property can be expressed in two equivalent ways:
1) for each object $C$ we have $\Sub(C) ≅ \Hom(C,Ω)$, and 2) for
every monic $B \monicto C$ there is exactly one map $χ_B:C→Ω$ making
the square below --- ``the Q-shaped diagram'' --- a pullback:
%
$$\diag{Omega-and-j}
$$
\item a local operator (also called a
``modality'', a ``Lawvere-Tierney topology'', or a ``topology'') is
a map $j:Ω→Ω$ obeying $j∘⊤=⊤$, $j∘j=j$ and $j∘∧=∧∘(j×j)$,
% (find-books "__cats/__cats.el" "mclarty")
% (find-mclartypage (+ 4 196) "21. Topologies")
\item a local operator $j$ induces a $j$-closure operator --- see
chapter 21 of \cite{McLarty} or chapter 5 of \cite{BellLST} ---, and
this $j$-closure operator can be seen as a map from each $\Sub(C)$
to itself. The closure of a subobject $i: B \monicto C$ is the
subobject $\ovl 1 : \ovl B \monicto C$ obtained by pullback in the
diagram below (``the rectangle''):
%
$$\diag{Omega-and-j-2}
$$
\end{itemize}
We will write the restriction of a local operator $j$ to $\Sub(1)$ as
$\sfJ(j)$ and we will say that a $j$ is ``associated to'' a $J$ when
$\sfJ(j) = J$.
\msk
There are two ways to ``understand'' the pullbacks above: the first
one is by doing the calculations formally and checking that everything
works, the second one is by checking some particular cases and
developing visual intuition from that.
% In the next sections I will refer to the two diagrams above as ``the
% Q-shaped diagram'' and ``the rectangle''.
\newpage
% ____ ____ __ _ _
% | _ \| __ ) ___ / _| ___ _ __ _ __ ___ __ _| | |_ _
% | |_) | _ \/ __| | |_ / _ \| '__| '_ ` _ \ / _` | | | | | |
% | __/| |_) \__ \ | _| (_) | | | | | | | | (_| | | | |_| |
% |_| |____/|___/ |_| \___/|_| |_| |_| |_|\__,_|_|_|\__, |
% |___/
%
% «pullbacks-formally» (to ".pullbacks-formally")
% (jonp 32 "pullbacks-formally")
% (joo "pullbacks-formally")
\subsection{Understanding the pullbacks formally}
\label{pullbacks-formally}
The calculations are routine if we know the right language, and if we
suppose --- without loss of generality --- that the monix $i:B\monicto
C$ is a ``canonical subobject'' in the sense that each $B(p)⊆C(p)$ and
each function $B(p\ton!q):B(p)→B(q)$ is a restriction of the
corresponding function $C(p\ton!q):C(p)→C(q)$.
We need some definitions:
%
$$\begin{array}{rcl}
1(p) &=& \{*\} \\
1(p\ton!q)(*) &=& * \\
Ω(p) &=& \Sub(↓p) \\
Ω(p\ton!q)(R) &=& R∧↓q \\%
%
[5pt]
%
⊤(p)(*) &=& ↓p \\
j(p)(R) &=& R^*∧↓p \\
χ_B(p)(R) &=& \setofst{r∈↓p}{C(p\ton!r)(c)∈B(r)} \\
\end{array}
$$
The first step is to check the five naturality conditions in the next
page --- we leave the rest to the reader. The main exercise is to
check that if the monic $i:B\monicto C$ is $i:P\monicto 1$ for a
truth-value $P$ then its closure is $i:\ovl P\monicto 1$ with $\ovl P$
being exactly $J(P)$, i.e., $P^*$.
\newpage
% «NTs-B-1» (to ".NTs-B-1")
%
%D diagram B->1
%D 2Dx 100 +30 +40 +35 +45
%D 2D 100 p Bp ---> 1p b |---> *p1
%D 2D | | | - -
%D 2D | | | | |
%D 2D | | | | v
%D 2D +25 v v v v *p2
%D 2D +8 q Bq --> 1q rb |--> *p3
%D 2D
%D 2D +20 B ----> 1
%D 2D
%D ren Bp Bq 1p 1q ==> B(p) B(q) \{*\} \{*\}
%D ren b rb *p1 *p2 *p3 ==> b B(p\ton!q)(b) * * *
%D
%D (( p q -> .plabel= l !
%D Bp 1p -> .plabel= a !
%D Bp Bq -> .plabel= l B(p\ton!q)
%D 1p 1q -> .plabel= r !
%D Bq 1q -> .plabel= a !
%D B 1 ->
%D
%D b *p1 |-> *p1 *p2 |->
%D b rb |-> rb *p3 |->
%D ))
%D enddiagram
%
% «NTs-B-C» (to ".NTs-B-C")
%
%D diagram B->C
%D 2Dx 100 +30 +40 +35 +55
%D 2D 100 p Bp ---> Cp b |---> cb
%D 2D | | | - -
%D 2D | | | | |
%D 2D | | | | v
%D 2D +25 v v v v rcb
%D 2D +8 q Bq ---> Cq rb |--> crb
%D 2D
%D 2D +20 B ----> C
%D 2D
%D ren Bp Bq Cp Cq ==> B(p) B(q) C(p) C(q)
%D ren b cb rcb rb crb ==> b b C(p\ton!q)(b) B(p\ton!q)(b) B(p\ton!q)(b)
%D
%D (( p q -> .plabel= l !
%D Bp Cp `-> .plabel= a ip
%D Bp Bq -> .plabel= l B(p\ton!q)
%D Cp Cq -> .plabel= r C(p\ton!q)
%D Bq Cq `-> .plabel= a iq
%D B C `-> .plabel= a i
%D
%D b cb |-> cb rcb |->
%D b rb |-> rb crb |->
%D ))
%D enddiagram
%
% «NTs-1-Om» (to ".NTs-1-Om")
%
%D diagram 1->Om
%D 2Dx 100 +30 +40 +35 +45
%D 2D 100 p 1p ---> Omp * |---> t*
%D 2D | | | - -
%D 2D | | | | |
%D 2D | | | | v
%D 2D +25 v v v v rt*
%D 2D +8 q 1q --> Omq r* |--> tr*
%D 2D
%D 2D +20 1 ----> Om
%D 2D
%D ren 1p 1q Omp Omq ==> \{*\} \{*\} \Sub(↓p) \Sub(↓q)
%D ren * t* rt* r* tr* ==> * ↓p ↓p∧↓q * ↓q
%D ren Om ==> Ω
%D
%D (( p q -> .plabel= l !
%D 1p Omp -> .plabel= a ⊤p
%D 1p 1q -> .plabel= l !
%D Omp Omq -> .plabel= r !
%D 1q Omq -> .plabel= a ⊤q
%D 1 Om -> .plabel= a ⊤
%D
%D * t* |-> t* rt* |->
%D * r* |-> r* tr* |->
%D ))
%D enddiagram
%
% «NTs-C-Om» (to ".NTs-C-Om")
%
%D diagram C->Om
%D 2Dx 100 +30 +40 +45 +95
%D 2D 100 p Cp ---> Omp c |---> chic
%D 2D | | | - -
%D 2D | | | | |
%D 2D | | | | v
%D 2D +25 v v v v rchic
%D 2D +8 q Cq ---> Omq rc |--> chirc
%D 2D
%D 2D +20 C ----> Om
%D 2D
%D ren Cp Omp ==> C(p) \Sub(↓p)
%D ren Cq Omq ==> C(q) \Sub(↓q)
%D ren C Om ==> C Ω
%D
%D ren c chic ==> c \setofst{r∈↓p}{C(p\ton!r)(c)∈B(r)}
%D ren rchic ==> \setofst{r∈↓p}{C(p\ton!r)(c)∈B(r)}∧↓q
%D ren rc chirc ==> C(p\ton!q)(c) \setofst{s∈↓q}{C(q\ton!s)(C(p\ton!q)(c))∈B(s)}
%D
%D (( p q -> .plabel= l !
%D Cp Omp -> .plabel= a χ_B(p)
%D Cp Cq -> .plabel= l C(p\ton!q)
%D Omp Omq -> .plabel= r Ω(p\ton!q)
%D Cq Omq -> .plabel= a χ_B(q)
%D C Om -> .plabel= a χ_B
%D
%D c chic |-> chic rchic |->
%D c rc |-> rc chirc |->
%D ))
%D enddiagram
%
% «NTs-Om-Om» (to ".NTs-Om-Om")
%
%D diagram Om->Om
%D 2Dx 100 +30 +40 +35 +45
%D 2D 100 p Sp1 --> Sp2 R |---> jR
%D 2D | | | - -
%D 2D | | | | |
%D 2D | | | | v
%D 2D +25 v v v v rjR
%D 2D +8 q Sq1 --> Sq2 rR |--> jrR
%D 2D
%D 2D +20 Om1 --> Om2
%D 2D
%D ren Sp1 Sp2 ==> \Sub(↓p) \Sub(↓p)
%D ren Sq1 Sq2 ==> \Sub(↓q) \Sub(↓q)
%D ren Om1 Om2 ==> Ω Ω
%D ren R jR rjR ==> R R^*∧↓p (R^*∧↓p)∧↓q
%D ren rR jrR ==> R∧↓q (R∧↓q)^*∧↓q
%D
%D (( p q -> .plabel= l !
%D Sp1 Sp2 -> .plabel= a j(p)
%D Sp1 Sq1 -> .plabel= l Ω(p\ton!q)
%D Sp2 Sq2 -> .plabel= r Ω(p\ton!q)
%D Sq1 Sq2 -> .plabel= a j(q)
%D Om1 Om2 -> .plabel= a j
%D
%D R jR |-> jR rjR |->
%D R rR |-> rR jrR |->
%D ))
%D enddiagram
%
% «fig:five-sqconds» (to ".fig:five-sqconds")
% (jonp 34 "fig:five-sqconds")
% (joo "fig:five-sqconds")
%
\pu
\begin{figure}[h!]
\centering
$\pu
\scalebox{0.9}{$
\begin{array}{l}
\diag{B->1} \\ \\
\diag{B->C} \\ \\
\diag{1->Om} \\ \\
\diag{C->Om} \\ \\
\diag{Om->Om} \\
\end{array}
$}
$
\caption{The five square conditions in the Q-shaped diagram}
\label{fig:five-sqconds}
\end{figure}
\newpage
% ____ ____ _ _ _
% | _ \| __ ) ___ __ _(_)___ _ _ __ _| | |_ _
% | |_) | _ \/ __| \ \ / / / __| | | |/ _` | | | | | |
% | __/| |_) \__ \ \ V /| \__ \ |_| | (_| | | | |_| |
% |_| |____/|___/ \_/ |_|___/\__,_|\__,_|_|_|\__, |
% |___/
%
% «pullbacks-visually» (to ".pullbacks-visually")
% (jonp 35 "pullbacks-visually")
% (joo "pullbacks-visually")
\subsection{Understanding the pullbacks visually}
\label{pullbacks-visually}
The best way to develop visual intuition about the $Ω$ and the $j$
associated to a $((P,A),Q)$ is to try to work out the details in some
particular cases --- I've chosen two, presented as execises below.
They both use the $((P,A),Q)$, the $Ω$ and the $j$ from
Figure \ref{fig:classifier-big}.
\msk
{\bf Exercise 1.} In the case
%
%D diagram Omega-and-j-exercise-1-Q
%D 2Dx 100 +30 +30
%D 2D 100 B --> 1
%D 2D | |
%D 2D v v
%D 2D +30 C --> Om1 --> Om2
%D 2D
%D ren B C ==> 11 33
%D ren Om1 Om2 ==> Ω Ω
%D
%D (( B 1 -> .plabel= a !
%D B C >-> .plabel= l i
%D 1 Om1 >-> .plabel= r ⊤
%D C Om1 -> .plabel= a χ_B
%D Om1 Om2 -> .plabel= a j
%D B relplace 7 7 \pbsymbol{7}
%D ))
%D enddiagram
%D
%D diagram Omega-and-j-exercise-1-rect
%D 2Dx 100 +30 +30
%D 2D 100 B ----------> 1
%D 2D | |
%D 2D v v
%D 2D +30 C --> Om1 --> Om2
%D 2D
%D ren B C ==> \ovl{11} 33
%D ren Om1 Om2 ==> Ω Ω
%D
%D (( B 1 -> .plabel= a !
%D B C >-> .plabel= l \ovl{i}
%D 1 Om2 >-> .plabel= r ⊤
%D C Om1 -> .plabel= a χ_B
%D Om1 Om2 -> .plabel= a j
%D B relplace 7 7 \pbsymbol{7}
%D ))
%D enddiagram
%D
\pu
$$
\diag{Omega-and-j-exercise-1-Q}
\qquad
\diag{Omega-and-j-exercise-1-rect}
$$
%
what is $χ_B$? And what is $\ovl{11}$?
\msk
{\bf Exercise 2.} In the case
%
%D diagram Omega-and-j-exercise-2-Q
%D 2Dx 100 +30 +30
%D 2D 100 B --> 1
%D 2D | |
%D 2D v v
%D 2D +30 C --> Om1 --> Om2
%D 2D
%D ren B C ==> 11 23
%D ren Om1 Om2 ==> Ω Ω
%D
%D (( B 1 -> .plabel= a !
%D B C >-> .plabel= l i
%D 1 Om1 >-> .plabel= r ⊤
%D C Om1 -> .plabel= a χ_B
%D Om1 Om2 -> .plabel= a j
%D B relplace 7 7 \pbsymbol{7}
%D ))
%D enddiagram
%D
%D diagram Omega-and-j-exercise-2-rect
%D 2Dx 100 +30 +30
%D 2D 100 B ----------> 1
%D 2D | |
%D 2D v v
%D 2D +30 C --> Om1 --> Om2
%D 2D
%D ren B C ==> \ovl{11} 23
%D ren Om1 Om2 ==> Ω Ω
%D
%D (( B 1 -> .plabel= a !
%D B C >-> .plabel= l \ovl{i}
%D 1 Om2 >-> .plabel= r ⊤
%D C Om1 -> .plabel= a χ_B
%D Om1 Om2 -> .plabel= a j
%D B relplace 7 7 \pbsymbol{7}
%D ))
%D enddiagram
%D
\pu
$$
\diag{Omega-and-j-exercise-2-Q}
\qquad
\diag{Omega-and-j-exercise-2-rect}
$$
%
what is $χ_B$? And what is $\ovl{11}$?
% (elep 6 "elephant-A2.1.3")
% (ele "elephant-A2.1.3")
% (elep 7 "elephant-A4.1.4")
% (ele "elephant-A4.1.4")
% (elep 8 "elephant-A4.1.5")
% (ele "elephant-A4.1.5")
% (ph1p 25 "topologies-as-partial-orders")
% (ph1 "topologies-as-partial-orders")
\directlua{tf_pop()}
% Local Variables:
% coding: utf-8-unix
% ee-tla: "joo"
% End: