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% (find-angg "LATEX/2010lcccs.tex")
% (find-dn4ex "edrx08.sty")
% (find-angg ".emacs.templates" "s2008a")
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2010lcccs.tex && latex 2010lcccs.tex"))
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2010lcccs.tex && pdflatex 2010lcccs.tex"))
% (eev "cd ~/LATEX/ && Scp 2010lcccs.{dvi,pdf} edrx@angg.twu.net:slow_html/LATEX/")
% (defun d () (interactive) (find-dvipage "~/LATEX/2010lcccs.dvi"))
% (find-dvipage "~/LATEX/2010lcccs.dvi")
% (find-pspage "~/LATEX/2010lcccs.ps")
% (find-pspage "~/LATEX/2010lcccs.pdf")
% (find-xpdfpage "~/LATEX/2010lcccs.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvipdf 2010lcccs.pdf 2010lcccs.dvi")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o 2010lcccs.ps 2010lcccs.dvi")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 600 -P pk -o 2010lcccs.ps 2010lcccs.dvi && ps2pdf 2010lcccs.ps 2010lcccs.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o tmp.ps tmp.dvi")
% (find-pspage "~/LATEX/tmp.ps")
% (ee-cp "~/LATEX/2010lcccs.pdf" (ee-twupfile "LATEX/2010lcccs.pdf") 'over)
% (ee-cp "~/LATEX/2010lcccs.pdf" (ee-twusfile "LATEX/2010lcccs.pdf") 'over)
% (find-twusfile "LATEX/" "2010lcccs")
% http://angg.twu.net/LATEX/2010lcccs.pdf
\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 2010lcccs.dnt
%*
% (eedn4-51-bounded)
%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")
Locally Cartesian Closed Categories (LCCCs)
\msk
Downcasings for the adjoints to change-of-base functors:
%:*=*{=}*
%:*∧*{∧}*
%:*⊃*{⊃}*
%D diagram LCCC-adjs-to-cob
%D 2Dx 100 +30 +30 +30 +35 +40
%D 2D 100 A0 ==> A1 B0 ==> B1 E0 ==> E1
%D 2D - - - - - -
%D 2D | <-> | | <-> | | <-> |
%D 2D v v v v v v
%D 2D +20 A2 <== A3 B2 <== B3 E2 <== E3
%D 2D - - - - - -
%D 2D | <-> | | <-> | | <-> |
%D 2D v v v v v v
%D 2D +20 A4 ==> A5 B4 ==> B5 E4 ==> E5
%D 2D
%D 2D +15 A6 |-> A7 B6 |-> B7 E6 |-> E7
%D 2D
%D 2D
%D 2D +25 C0 ==> C1 D0 ==> D1 F0 ==> F1
%D 2D - - - - - -
%D 2D | <-> | | <-> | | <-> |
%D 2D v v v v v v
%D 2D +20 C2 <== C3 D2 <== D3 F2 <== F3
%D 2D - - - - - -
%D 2D | <-> | | <-> | | <-> |
%D 2D v v v v v v
%D 2D +20 C4 ==> C5 D4 ==> D5 F4 ==> F5
%D 2D
%D 2D +15 C6 |-> C7 D6 |-> D7 F6 |-> F7
%D 2D
%D (( A0 .tex= c A1 .tex= b,c
%D A2 .tex= d A3 .tex= d
%D A4 .tex= e A5 .tex= b|->e
%D A6 .tex= a,b A7 .tex= a
%D @ 0 @ 1 => @ 2 @ 3 <= @ 4 @ 5 => @ 6 @ 7 |-> .plabel= a \pi
%D @ 0 @ 2 |-> @ 1 @ 3 |-> @ 0 @ 3 harrownodes nil 20 nil <->
%D @ 2 @ 4 |-> @ 3 @ 5 |-> @ 2 @ 5 harrownodes nil 20 nil <->
%D ))
%D (( B0 .tex= p B1 .tex= (b=b'),p
%D B2 .tex= \ov{q} B3 .tex= q
%D B4 .tex= r B5 .tex= (b=b')|->r
%D B6 .tex= a,b B7 .tex= a,b,b'
%D @ 0 @ 1 => @ 2 @ 3 <= @ 4 @ 5 => @ 6 @ 7 |-> .plabel= a \dd
%D @ 0 @ 2 |-> @ 1 @ 3 |-> @ 0 @ 3 harrownodes nil 20 nil <->
%D @ 2 @ 4 |-> @ 3 @ 5 |-> @ 2 @ 5 harrownodes nil 20 nil <->
%D ))
%D (( C0 .tex= P C1 .tex= ýb.P
%D C2 .tex= Q C3 .tex= Q
%D C4 .tex= R C5 .tex= ýb.R
%D C6 .tex= a,b C7 .tex= a
%D @ 0 @ 1 => @ 2 @ 3 <= @ 4 @ 5 => @ 6 @ 7 |-> .plabel= a \pi
%D @ 0 @ 2 |-> @ 1 @ 3 |-> @ 0 @ 3 harrownodes nil 20 nil <->
%D @ 2 @ 4 |-> @ 3 @ 5 |-> @ 2 @ 5 harrownodes nil 20 nil <->
%D ))
%D (( D0 .tex= P D1 .tex= (b=b')∧P
%D D2 .tex= \ov{Q} D3 .tex= Q
%D D4 .tex= R D5 .tex= (b=b')⊃R
%D D6 .tex= a,b D7 .tex= a,b,b'
%D @ 0 @ 1 => @ 2 @ 3 <= @ 4 @ 5 => @ 6 @ 7 |-> .plabel= a \dd
%D @ 0 @ 2 |-> @ 1 @ 3 |-> @ 0 @ 3 harrownodes nil 20 nil <->
%D @ 2 @ 4 |-> @ 3 @ 5 |-> @ 2 @ 5 harrownodes nil 20 nil <->
%D ))
%D (( E0 .tex= p_a E1 .tex= a,(b=fa),p_a
%D E2 .tex= q_{fa} E3 .tex= q_b
%D E4 .tex= r_a E5 .tex= (a,(b=fa))|->r_a
%D E6 .tex= a E7 .tex= b
%D @ 0 @ 1 => @ 2 @ 3 <= @ 4 @ 5 => @ 6 @ 7 |-> .plabel= a f
%D @ 0 @ 2 |-> @ 1 @ 3 |-> @ 0 @ 3 harrownodes nil 20 nil <->
%D @ 2 @ 4 |-> @ 3 @ 5 |-> @ 2 @ 5 harrownodes nil 20 nil <->
%D ))
%D (( F0 .tex= Pa F1 .tex= Îa.b=fa∧Pa
%D F2 .tex= Qfa F3 .tex= Qb
%D F4 .tex= Ra F5 .tex= ýa.b=fa⊃Ra
%D F6 .tex= a F7 .tex= b
%D @ 0 @ 1 => @ 2 @ 3 <= @ 4 @ 5 => @ 6 @ 7 |-> .plabel= a f
%D @ 0 @ 2 |-> @ 1 @ 3 |-> @ 0 @ 3 harrownodes nil 20 nil <->
%D @ 2 @ 4 |-> @ 3 @ 5 |-> @ 2 @ 5 harrownodes nil 20 nil <->
%D ))
%D ((
%D
%D ))
%D enddiagram
%D
$$\diag{LCCC-adjs-to-cob}$$
\msk
Other operations:
%D diagram LCCC-CCCs
%D 2Dx 100 +25 +25 +15 +20 +25 +40 +35
%D 2D 100 p0 t0 e0 e1 pb0 pb1
%D 2D
%D 2D +25 p1 p2 p3 t1 e2 e3 pb2 pb3
%D 2D
%D 2D +20 P0 T0 E0 E1
%D 2D
%D 2D +25 P1 P2 P3 T1 E2 E3
%D 2D
%D (( p0 .tex= b p1 .tex= c p2 .tex= c,d p3 .tex= d
%D p0 p1 |->
%D p0 p2 |->
%D p0 p3 |->
%D p1 p2 <-|
%D p2 p3 |->
%D ))
%D (( t0 .tex= b t1 .tex= * |->
%D ))
%D (( e0 .tex= b,c e1 .tex= b
%D e2 .tex= d e3 .tex= c|->d
%D e0 e1 <=
%D e0 e2 |->
%D e1 e3 |->
%D e0 e3 harrownodes nil 20 nil <->
%D e2 e3 =>
%D ))
%D (( P0 .tex= P P1 .tex= Q P2 .tex= Q∧R P3 .tex= R
%D P0 P1 |->
%D P0 P2 |->
%D P0 P3 |->
%D P1 P2 <-|
%D P2 P3 |->
%D ))
%D (( T0 .tex= P T1 .tex= § |->
%D ))
%D (( E0 .tex= P∧Q E1 .tex= P
%D E2 .tex= R E3 .tex= Q⊃R
%D E0 E1 <=
%D E0 E2 |->
%D E1 E3 |->
%D E0 E3 harrownodes nil 20 nil <->
%D E2 E3 =>
%D ))
%D (( pb0 .tex= a,b,(fa=gb) pb1 .tex= b
%D pb2 .tex= a pb3 .tex= c
%D @ 0 @ 1 |->
%D @ 0 @ 2 |-> @ 1 @ 3 |-> .plabel= r g
%D @ 2 @ 3 |-> .plabel= a f
%D @ 0 relplace 8 8 \pbsymbol{7}
%D ))
%D enddiagram
%D
$$\diag{LCCC-CCCs}$$
%*
\end{document}
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