Warning: this is an htmlized version!
The original is across this link,
and the conversion rules are here.
% (find-angg "LATEX/2008dclosed.tex")
% (find-dn4ex "edrx08.sty")
% (find-angg ".emacs.templates" "s2008a")
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2008dclosed.tex && latex    2008dclosed.tex"))
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2008dclosed.tex && pdflatex 2008dclosed.tex"))
% (eev "cd ~/LATEX/ && Scp 2008dclosed.{dvi,pdf} edrx@angg.twu.net:slow_html/LATEX/")
% (find-dvipage "~/LATEX/2008dclosed.dvi")
% (find-pspage  "~/LATEX/2008dclosed.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o 2008dclosed.ps 2008dclosed.dvi")
% (find-pspage  "~/LATEX/2008dclosed.ps")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o tmp.ps tmp.dvi")
% (find-pspage  "~/LATEX/tmp.ps")
% (ee-cp "~/LATEX/2008dclosed.pdf" (ee-twupfile "LATEX/2008dclosed.pdf") 'over)
% (ee-cp "~/LATEX/2008dclosed.pdf" (ee-twusfile "LATEX/2008dclosed.pdf") 'over)

% «.why-slides»			(to "why-slides")
% «.mod-induces-dclosed»	(to "mod-induces-dclosed")
% «.mod-induces-dclosed-2»	(to "mod-induces-dclosed-2")
% «.dncing-dense-closed»	(to "dncing-dense-closed")
% «.dncing-dense-closed-2»	(to "dncing-dense-closed-2")
% «.etc»			(to "etc")


\documentclass[oneside]{book}
\usepackage[latin1]{inputenc}
\usepackage{edrx08}       % (find-dn4ex "edrx08.sty")
%L process "edrx08.sty"  -- (find-dn4ex "edrx08.sty")
\input edrxheadfoot.tex   % (find-dn4ex "edrxheadfoot.tex")
\begin{document}

\input 2008dclosed.dnt

%*
% (eedn4-51-bounded)

Notes about factorization systems -

esp.\ the dense/closed factorizations for monics.

\bsk

Index of the slides:
\msk
% To update the list of slides uncomment this line:
\makelos{tmp.los}
% then rerun LaTeX on this file, and insert the contents of "tmp.los"
% below, by hand (i.e., with "insert-file"):
% (find-fline "tmp.los")
% (insert-file "tmp.los")
\tocline {Why slides} {2}
\tocline {A modality induces a dense/close factorization} {3}
\tocline {A modality induces a dense/close factorization (2)} {4}
\tocline {Downcasing dense and closed maps} {5}
\tocline {Downcasing dense and closed maps (2)} {6}
\tocline {etc} {7}

%:*>->*\monicto *
%:*<-<*\monicot *
%L forths[">.>"] = function () pusharrow(" >.>") end
%L forths["<.<"] = function () pusharrow(" <.<") end
%L forths["`.>"] = function () pusharrow("^{ (}.>") end
%L forths["|-"] = function () pusharrow("|-") end
%L forths["-|"] = function () pusharrow("-|") end

\def\j{j}
\def\sm#1{\begin{smallmatrix}#1\end{smallmatrix}}


\newpage
% --------------------
% «why-slides»  (to ".why-slides")
% (s "Why slides" "why-slides")
\myslide {Why slides} {why-slides}

% (find-es "tex" "smash")
\def\mysmash#1{{\setbox0\hbox{#1}%
  \wd0=0pt\ht0=0pt\dp0=0pt%
  \box0}}
\def\tl#1{\begin{tabular}{l}#1\end{tabular}}

%D diagram ??
%D 2Dx     100      +60
%D 2D  100 \boxrps  \boxcts
%D 2D	            
%D 2D  +40 \boxssl  \boxlpp
%D 2D	            
%D 2D  +40 \boxips  \boxstu
%D 2D
%D (( \boxrps  \boxcts
%D    \boxssl  \boxlpp
%D    \boxips  \boxstu
%D    @ 0 @ 1 ->
%D    @ 0 @ 3 -> @ 2 @ 1 ->
%D    @ 2 @ 3 ->
%D    @ 2 @ 5 -> @ 4 @ 3 ->
%D    @ 4 @ 5 ->
%D ))
%D enddiagram
%D
$$
  \def\boxcts{\tl{Category \\ Theorists}}
  \def\boxlpp{\tl{local alge\mysmash{braic} \\ geometers / \\ logicians / \\ CS people}}
  \def\boxstu{\tl{students}}
  \def\boxrps{\tl{``real'' \\ papers}}
  \def\boxssl{\tl{seminars, \\ slides}}
  \def\boxips{\tl{introductory \\ papers}}
  \diag{??}
$$

moral proofs, intuition, archetypal examples

new language, with translation (not as formal as C/H)

private devices

lifting and generalization

Several kinds of brownie points (cite Simmons)

Translation of notations in several articles 


A new language, vs.\ a new truth.

Draw the mnemonics for 

Sometimes I think that this may look like an autistic exercise - too
much energy spent on just rephrasing two well-known papers into a new
notation.

{\sl How does one work the standard, ``algebraic'' notation?} I've
spent 

Even after working with it for years, the standard, ``algebraic''
notation, 

However, the downcased notation reflects exactly how I think about
certain problems, and I still find - even after many years - the usual
(I'll call it ``standard'') algebraic notation hard to follow.

This is probably because I don't know the tricks.

What are the mnemonics/skeletons?

Social effects:

few people around me know category theory

some people to whom I've shown this have found it very nice (maybe
they were just being polite?)

there's a gap between not knowing and knowing CT

\bsk

Truth vs. translation

Making things obvious, tautological; zooming into proofs

I will try to describe it in linguistical terms. 

Native speaker

\msk

A new language, vs.\ a new truth.

Only one real ``theorem'' --- about filter-powers. It needs sheaves.

The big metatheorems ahead involve parametricity (and polymorphism and hyperdoctrines).

Another possibility: Property-Like Structures.

\msk

I do not want the brownie points now [Simmons].

Sheaves: I learned a lot from Simmons course notes.

Actually there are several ways to get brownie points - seminars,
talks in conferences, introductory papers, 

I'm looking for coauthors (and energy)

Translate the notation in several important books

\msk



Barr: *-Autonomous Cats

Bierman/dePaiva: S4

Cheng: Mathematics, Morally

Corfield: [Towards a Philosophy of Real Mathematics]

Freyd: Algebraic categories

Jacobs: Tijolão

Jacobs: Comprehension Cats

Joyal/Street: On the Geometry of Tensor Calculus

Kelly/Lack: On Property-Like Structures

Kock: A simple axiomatics for differentiation

Lawvere (easy book): external/internal view

MacLane: CWM

Pitts: Polymorphism is Set-Theoretic, constructively

Reyes/Zolfaghari: Topos-theoretic approaches to modality

Reynolds: Polymorphism is notCategorical Semantics for Higher Order Polymorphic Lambda Calculus Set-Theoretic

Seely: Hyperdoctrines

Seely: Categorical Semantics for Higher Order Polymorphic Lambda Calculus

Seely: Differential Cats

Wadler: The Girard/Reynods iso

Wadler: Theorems for Free







\newpage
% --------------------
% «mod-induces-dclosed»  (to ".mod-induces-dclosed")
% (s "A modality induces a dense/close factorization" "mod-induces-dclosed")
\myslide {A modality induces a dense/close factorization} {mod-induces-dclosed}

Any map $j: \to $ induces ``the right half'' of a factorization

for the monics of a topos, in a way that is stable by pullbacks:

any diagram $A \to B \monicot P$ induces the bigger diagram below,

%D diagram mod-ind-fact
%D 2Dx     100 +20  +20 +20   +40 +20 +20
%D 2D  100     AP       BP    1
%D 2D
%D 2D  +30 AP*      BP*           1*
%D 2D
%D 2D  +30     A        B            *
%D 2D
%D (( AP  .tex= A×_BP   BP  .tex= P
%D    AP* .tex= A×_BP^* BP* .tex= P^*   1* .tex= 1
%D    *  .tex= 
%D    BP* .tex= \j_BP
%D    AP* .tex= \sm{\j_A(A×_BP)\,\cong\\A×_B(\j_BP)}
%D ))
%D (( AP BP -> BP 1 ->
%D    AP AP* >.> AP* A >-> AP A >->
%D    BP BP* >.> BP* B >-> BP B >->
%D    AP* BP* -> BP* 1* ->
%D    A B ->  .plabel= b f B  -> .plabel= b p  * -> .plabel= b j
%D    1  ->  1* * ->
%D    AP _| BP _|
%D    AP* relplace 15 6 \pbsymbol{7}
%D    BP* relplace 11 5 \pbsymbol{7}
%D ))
%D enddiagram
%D
$$\diag{mod-ind-fact}$$

where the `$\j_BP$'s are constructed as this:

%:  P>->B
%:  ------
%:  p:B->  j:->
%:  --------------
%:    (p;j):B->
%:    ----------
%:    \j_BP>->B
%:
%:    ^jBP-construction
%:
$$\ded{jBP-construction}$$

$\j_A(A×_BP) \monicto A$ is classified by $(f;p);j$ and

$A×_B(\j_BP) \monicto A$ is classified by $f;(p;j)$,

so they are isomorphic.

\msk

If $j$ is ``inflationary'' then the dotted monics above exist.

This becomes easier to understand if we use a more logical language:
%
%D diagram P-to-jBP
%D 2Dx     100  +35   +30  +45
%D 2D  100 A0   A1    a0   a1
%D 2D
%D 2D  +30 A2   A3    a2   a3
%D 2D
%D (( A0 .tex= \j_BP×_BP  A1 .tex= P
%D    A2 .tex= \j_BP      A3 .tex= B
%D    A0 A1 <->
%D    A0 A2 >-> A1 A2 >.> A1 A3 >->
%D    A2 A3 >->
%D ))
%D (( a0 .tex= \sst{b}{P^*{∧}P}  a1 .tex= \sst{b}{P}
%D    a2 .tex= \sst{b}{P^*}    a3 .tex= B
%D    a0 a1 <->
%D    a0 a2 >-> a1 a2 >.> a1 a3 >->
%D    a2 a3 >->
%D ))
%D enddiagram
%:
%:    P|-P^*
%:  ==========
%:  P∧P|-P^*∧P
%:  ==========
%:   P|-P^*∧P
%:
%:   ^P-to-jBP
%:
$$\cdiag{P-to-jBP} \qquad \cded{P-to-jBP}$$

\msk

By one of the standard lemmas on glueing pullbacks,

the square with the dotted monics in the top diagram is a pullback.



\newpage
% --------------------
% «mod-induces-dclosed-2»  (to ".mod-induces-dclosed-2")
% (s "A modality induces a dense/close factorization (2)" "mod-induces-dclosed-2")
\myslide {A modality induces a dense/close factorization (2)} {mod-induces-dclosed-2}

Any inflationary map $j: \to $ induces a ``factorization''

for the monics of a topos, in a way that is stable by pullbacks...

any diagram $A \to B \monicot P$ induces the diagram below,

%D diagram mod-ind-fact-2
%D 2Dx     100 +20  +20 +20   +40 +20 +20
%D 2D  100     AP       BP    1
%D 2D
%D 2D  +30     AP*      BP*           1*
%D 2D
%D 2D  +30     A        B            *
%D 2D
%D (( AP  .tex= A×_BP   BP  .tex= P
%D    AP* .tex= A×_BP^* BP* .tex= P^*   1* .tex= 1
%D    *  .tex= 
%D    BP* .tex= \j_BP
%D    AP* .tex= \sm{\j_A(A×_BP)\,\cong\\A×_B(\j_BP)}
%D ))
%D (( AP BP -> BP 1 ->
%D    AP AP* >-> AP* A >->
%D    BP BP* >-> BP* B >->
%D    AP* BP* -> BP* 1* ->
%D    A B ->  .plabel= b f B  -> .plabel= b p  * -> .plabel= b j
%D    1  ->  .plabel= m \hbox{\phantom{p}} 1* * ->
%D    AP _| BP _|
%D    AP* relplace 14 7 \pbsymbol{7}
%D    BP* relplace 7 7 \pbsymbol{7}
%D ))
%D enddiagram
%D
$$\diag{mod-ind-fact-2}$$

In the more logical notation, this becomes:

%D diagram mod-ind-fact-3
%D 2Dx     100 +20      +50   +40 +20 +20
%D 2D  100     AP       BP    1
%D 2D
%D 2D  +30     AP*      BP*           1*
%D 2D
%D 2D  +30     A        B            *
%D 2D
%D (( AP  .tex= \sst{a}{P(fa)}   BP  .tex= \sst{b}{P(b)}
%D    1* .tex= 1
%D    *  .tex= 
%D    BP* .tex= \sst{a}{P^*(b)}
%D    AP* .tex= \sm{\sst{a}{(Pf)^*(a)}\\\cong\,\sst{a}{P^*(fa)}}
%D ))
%D (( AP BP -> BP 1 ->
%D    AP AP* >-> AP* A >->
%D    BP BP* >-> BP* B >->
%D    AP* BP* -> BP* 1* ->
%D    A B ->  .plabel= b f B  -> .plabel= b p  * -> .plabel= b j
%D    1  ->  .plabel= m \hbox{\phantom{p}} 1* * ->
%D    AP  relplace  8  8 \pbsymbol{7}
%D    BP  relplace  8  8 \pbsymbol{7}
%D    AP* relplace 14 10 \pbsymbol{7}
%D    BP* relplace  9  8 \pbsymbol{7}
%D ))
%D enddiagram
%D
$$\diag{mod-ind-fact-3}$$

We will usually omit the parameter - as in `$P(fa)$' $\mto$ `$P$' -

and the `1's and `$$'s when we downcase this.

The case where $A \to B$ is a monic ($\equiv \; b|_Q \mto b$)

is especially interesting - then the left column is formed

by adding `$∧Q$'s.


%D diagram mod-ind-fact-4
%D 2Dx     100  +35    +40  +35
%D 2D  100 a0   a1     b0   b1
%D 2D
%D 2D  +30 a2   a3     b2   b3
%D 2D
%D 2D  +30 a4   a5     b4   b5
%D 2D
%D (( a0 .tex= a|_P      a1 .tex= b|_P
%D    a2 .tex= a|_{P^*}  a3 .tex= b|_{P^*}
%D    a4 .tex= a         a5 .tex= b
%D    a0 a1 `->
%D    a0 a2 `-> a1 a3 `->
%D    a2 a3 `->
%D    a2 a4 `-> a3 a5 `->
%D    a4 a5 `->
%D    a0 _| a2 _|
%D ))
%D (( b0 .tex= b|_{P∧Q}    b1 .tex= b|_P
%D    b2 .tex= b|_{P^*{∧}Q}  b3 .tex= b|_{P^*}
%D    b4 .tex= b|_{Q}      b5 .tex= b
%D    b0 b1 `->
%D    b0 b2 `-> b1 b3 `->
%D    b2 b3 `->
%D    b2 b4 `-> b3 b5 `->
%D    b4 b5 `->
%D    b0 _| b2 _|
%D ))
%D enddiagram
%D
$$\diag{mod-ind-fact-4}$$

{\bf Crucial fact:}

every dense map is of the form $b|_{P∧Q} \ito b|_{P^*{∧}Q}$, and

every closed map is of the form $b|_{P^*{∧}Q} \ito b|_Q$!


\newpage
% --------------------
% «dncing-dense-closed»  (to ".dncing-dense-closed")
% (s "Downcasing dense and closed maps" "dncing-dense-closed")
\myslide {Downcasing dense and closed maps} {dncing-dense-closed}

% (find-LATEX "2008hyp.tex" "adj-functors-as-seq-rules")

%D diagram foo
%D 2Dx       100      +60
%D 2D  100  P∧Q       R
%D 2D	     -   |b>  -
%D 2D	     |        |
%D 2D	     v   <#|  v
%D 2D  +30 P^*∧Q      S
%D 2D	     -   <b|  -
%D 2D	     |        |
%D 2D	     v   |#>  v
%D 2D  +30   Q        T
%D 2D
%D (( P∧Q R   P^*∧Q S   Q T
%D    @ 0 @ 2 |-
%D    @ 1 @ 3 |-
%D    @ 2 @ 4 |-
%D    @ 3 @ 5 |-
%D    @ 0 @ 3 harrownodes nil 20 nil -> sl^ .plabel= a R:=P∧Q,\;S:=P^*∧Q
%D    @ 0 @ 3 harrownodes nil 20 nil <- sl_ .plabel= b P':=R,\;Q':=S
%D    @ 2 @ 5 harrownodes nil 20 nil -> sl^ .plabel= a S:=P^*∧Q,\;T:=Q
%D    @ 2 @ 5 harrownodes nil 20 nil <- sl_ .plabel= b P':=S,\;Q':=T
%D ))
%D enddiagram

$$\diag{foo}$$

\widemtos

The composites

$(P∧Q \vdash P^*∧Q) \mto (R \vdash S) \mto (P'∧Q' \vdash P'^*∧Q')$

and

$(P^*∧Q \vdash Q) \mto (S \vdash T) \mto (P'^*∧Q' \vdash Q')$

are identity maps:

\ssk

in the first one,

$(P'∧Q' \vdash P'^*∧Q') =$

$(R∧S \vdash R^*∧S) =$

$((P∧Q)∧(P^*∧Q) \vdash (P∧Q)^*∧(P^*∧Q)) =$

$(P∧Q \vdash P^*∧Q)$;

\ssk

in the second one,

$(P'^*∧Q' \vdash Q') =$

$(S^*∧T \vdash T) =$

$((P^*∧Q)^*∧Q \vdash Q) =$

$(P^*∧Q \vdash Q)$.


\newpage
% --------------------
% «dncing-dense-closed-2»  (to ".dncing-dense-closed-2")
% (s "Downcasing dense and closed maps (2)" "dncing-dense-closed-2")
\myslide {Downcasing dense and closed maps (2)} {dncing-dense-closed-2}

%D diagram foo2
%D 2Dx       100      +60
%D 2D  100  P∧Q       R
%D 2D	     -   |b>  -
%D 2D	     |        |
%D 2D	     v   <#|  v
%D 2D  +30 P^*∧Q      S
%D 2D	     -   <b|  -
%D 2D	     |        |
%D 2D	     v   |#>  v
%D 2D  +30   Q        T
%D 2D
%D (( P∧Q R   P^*∧Q S   Q T
%D    @ 0 @ 2 |-
%D    @ 1 @ 3 |-
%D    @ 2 @ 4 |-
%D    @ 3 @ 5 |-
%D    @ 0 @ 3 harrownodes nil 20 nil -> sl^ .plabel= a R':=P∧Q,\;S':=P^*∧Q
%D    @ 0 @ 3 harrownodes nil 20 nil <- sl_ .plabel= b P:=R,\;Q:=S
%D    @ 2 @ 5 harrownodes nil 20 nil -> sl^ .plabel= a S':=P^*∧Q,\;T':=Q
%D    @ 2 @ 5 harrownodes nil 20 nil <- sl_ .plabel= b P:=S,\;Q:=T
%D ))
%D enddiagram

$$\diag{foo2}$$

The other two composites,

$(R \vdash S) \mto (P∧Q \vdash P^*∧Q) \mto (R' \vdash S')$

and 

$(S \vdash T) \mto (P^*∧Q \vdash Q) \mto (S' \vdash T')$,

should be identities iff $R \vdash S$ is dense and $S \vdash T$ is closed...

\ssk

Let's check. In the first one,

$(R' \vdash S') =$

$(P∧Q \vdash P^*∧Q) =$

$(R∧S \vdash R^*∧S) =$

$(R \vdash R^*∧S)$,

\ssk

and in the second one,

$(S' \vdash T') =$

$(P^*∧Q \vdash Q) =$

$(S^*∧T \vdash T)$;

\ssk

so what we need to prove is:

$R \vdash S$ is dense iff $S = R^*∧S$,

$S \vdash T$ is closed iff $S = S^*∧T$.

As $R^*∧S \vdash S$ and $S^*∧T \vdash T$, we just need to prove

$R \vdash S$ is dense iff $S \vdash R^*$,

$S \vdash T$ is closed iff $S \vdash S^*∧T$.




\newpage
% --------------------
% «etc»  (to ".etc")
% (s "etc" "etc")
\myslide {etc} {etc}

%:*P**P^**
%:*Q**Q^**

\def\arr#1#2{\begin{array}{#1}#2\end{array}}

%L -- (find-dn4file "dednat4.lua")
%L forths[".dp="] = function ()
%L     ds[1].placement = getword()
%L     ds[1].label = "(d)"
%L   end
%L forths[".cp="] = function ()
%L     ds[1].placement = getword()
%L     ds[1].label = "(c)"
%L   end


%D diagram perm1
%D 2Dx     100         +35       +35     +40     +50     +45
%D 2D  100 a|_{P}     a|_{P*}    a     A×_BC  A×_Bj_BC   A
%D 2D	                                                  
%D 2D  +30 b|_{P}     b|_{P*}    b       C      j_BC     B
%D 2D
%D (( a|_{P} a|_{P*} a
%D    b|_{P} b|_{P*} b
%D    @ 0 @ 1 `-> .dp= a  @ 1 @ 2 `-> .cp= a
%D    @ 3 @ 4 `-> .dp= a  @ 4 @ 5 `-> .cp= a
%D    @ 0 @ 3 |-> @ 1 @ 4 |-> @ 2 @ 5 |->
%D    @ 0 _| @ 1 _|
%D ))
%D (( A×_BC A×_Bj_BC A
%D      C     j_BC   B
%D    A×_Bj_BC .tex= \arr{c}{j_A{A×_BC}\\{\cong}A×_Bj_BC}
%D    @ 0 @ 1 >-> .dp= a  @ 1 @ 2 >-> .cp= a
%D    @ 3 @ 4 >-> .dp= a  @ 4 @ 5 >-> .cp= a
%D    @ 0 @ 3 -> @ 1 @ 4 -> @ 2 @ 5 ->
%D    @ 0 _| @ 1 _|
%D ))
%D enddiagram
%D
$$\diag{perm1}$$

%D diagram permanence2
%D 2Dx     100        +40         +35        +35         +45         +35
%D 2D  100 a|_{P∧Q}   a|_{P*∧Q}   a|_{Q}     A×_BC       j_C(A×_BC)  C
%D 2D	                                                
%D 2D  +30 a|_{P∧Q*}  a|_{P*∧Q*}  a|_{Q*}    j_B(A×_BC)  M           j_AC
%D 2D	                                                
%D 2D  +30 a|_{P}     a|_{P*}     a          B           j_AB        A
%D 2D
%D (( a|_{P∧Q}   a|_{P*∧Q}   a|_{Q}      
%D    a|_{P∧Q*}  a|_{P*∧Q*}  a|_{Q*}
%D    a|_{P}     a|_{P*}     a
%D    @ 0 @ 1 `-> .dp= a  @ 1 @ 2 `-> .cp= a
%D    @ 0 @ 3 `-> .dp= l  @ 1 @ 4 `-> .dp= l  @ 2 @ 5 `-> .dp= l
%D    @ 3 @ 4 `-> .dp= a  @ 4 @ 5 `-> .cp= a
%D    @ 3 @ 6 `-> .cp= l  @ 4 @ 7 `-> .cp= l  @ 5 @ 8 `-> .cp= l
%D    @ 6 @ 7 `-> .dp= a  @ 7 @ 8 `-> .cp= a
%D    @ 0 _| @ 1 _| @ 3 _| @ 4 _|
%D ))
%D (( A×_BC       j_C(A×_BC)  C
%D    j_B(A×_BC)  M           j_AC
%D    B           j_AB        A
%D    @ 0 @ 1 >-> .dp= a  @ 1 @ 2 >-> .cp= a
%D    @ 0 @ 3 >-> .dp= l  @ 1 @ 4 >-> .dp= l  @ 2 @ 5 >-> .dp= l
%D    @ 3 @ 4 >-> .dp= a  @ 4 @ 5 >-> .cp= a
%D    @ 3 @ 6 >-> .cp= l  @ 4 @ 7 >-> .cp= l  @ 5 @ 8 >-> .cp= l
%D    @ 6 @ 7 >-> .dp= a  @ 7 @ 8 >-> .cp= a
%D    @ 0 _| @ 1 _| @ 3 _| @ 4 _|
%D ))
%D enddiagram
%D
$$\diag{permanence2}$$

%D diagram permanence3
%D 2Dx     100        +35         +35    +35    +35    +35
%D 2D  100 a|_{P∧Q}                      C	       
%D 2D	                                                
%D 2D  +30 a|_{P∧Q*}  a|_{P*∧Q*}         j_BC  j_AC     
%D 2D	                                                
%D 2D  +30 a|_{P}     a|_{P*}     a      B     j_AB    A
%D 2D
%D (( a|_{P∧Q}                
%D    a|_{P∧Q*}  a|_{P*∧Q*}   
%D    a|_{P}     a|_{P*}     a
%D    @ 0 @ 1 `-> .dp= l  @ 0 @ 2 `-> .dp= r
%D    @ 1 @ 2 `-> .dp= b
%D    @ 1 @ 3 `-> .cp= l @ 2 @ 4 `-> .cp= l    @ 2 @ 5 `-> .cp= r
%D    @ 3 @ 4 `-> .dp= b @ 4 @ 5 `-> .cp= b
%D ))
%D (( C	   
%D    j_BC j_AC  
%D    B    j_AB A
%D    @ 0 @ 1 >-> .dp= l  @ 0 @ 2 >-> .dp= r
%D    @ 1 @ 2 >-> .dp= b
%D    @ 1 @ 3 >-> .cp= l @ 2 @ 4 >-> .cp= l    @ 2 @ 5 >-> .cp= r
%D    @ 3 @ 4 >-> .dp= b @ 4 @ 5 >-> .cp= b
%D ))
%D enddiagram
%D
$$\diag{permanence3}$$


%D diagram permanence4
%D 2Dx     100    +35      +35     +25   +0  +35 +35  +25
%D 2D  100 a|_{P} a|_{R}                 B   D
%D 2D
%D 2D  +30 a|_{Q} a|_{Q∨R}               C  C∨D
%D 2D
%D 2D  +25                a|_{P*}  a            j_AB  A
%D 2D
%D (( a|_{P} a|_{R}
%D    a|_{Q} a|_{Q∨R}
%D                    a|_{P*} a
%D    @ 0 @ 1 `-> .dp= a
%D    @ 0 @ 2 `-> .dp= l  @ 1 @ 3 `-> .dp= l  @ 1 @ 4 `-> .dp= r
%D    @ 2 @ 3 `-> .dp= a
%D    @ 2 @ 4 `-> .dp= b  @ 3 @ 4 `-> .dp= m  @ 4 @ 5 `-> .cp= a
%D ))
%D (( B D
%D    C C∨D
%D          j_AB A
%D    @ 0 @ 1 >-> .dp= a
%D    @ 0 @ 2 >-> .dp= l  @ 1 @ 3 >-> .dp= l  @ 1 @ 4 >-> .dp= r
%D    @ 2 @ 3 >-> .dp= a
%D    @ 2 @ 4 >-> .dp= b  @ 3 @ 4 >-> .dp= m  @ 4 @ 5 >-> .cp= a
%D ))
%D enddiagram
%D
$$\diag{permanence4}$$

\def\calU{{\mathcal{U}}}
\def\calV{{\mathcal{V}}}
\def\calW{{\mathcal{W}}}

An open set $U$ is above and to the right of a point $\cc$ when $\cc  U$. 

An open set $U$ is straight above a point $\aa$ when $\aa$ generates $U$.

A cover $\calU$ is above and to right of $W$ when $\bigcup \calU = W$.

A cover $\calU$ is straight above $U$ when $\bigcup \calU = U$.

% $\sst{a}{P(a)}$

%*




\end{document}

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