% (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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