% (find-angg "LATEX/2010sheaves.tex")
% (find-dn4ex "edrx08.sty")
% (find-angg ".emacs.templates" "s2008a")
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2010sheaves.tex && latex    2010sheaves.tex"))
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2010sheaves.tex && pdflatex 2010sheaves.tex"))
% (eev "cd ~/LATEX/ && Scp 2010sheaves.{dvi,pdf} edrx@angg.twu.net:slow_html/LATEX/")
% (defun d () (interactive) (find-dvipage "~/LATEX/2010sheaves.dvi"))
% (find-dvipage "~/LATEX/2010sheaves.dvi")
% (find-pspage  "~/LATEX/2010sheaves.ps")
% (find-pspage  "~/LATEX/2010sheaves.pdf")
% (find-xpdfpage "~/LATEX/2010sheaves.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvipdf         2010sheaves.dvi 2010sheaves.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o 2010sheaves.ps 2010sheaves.dvi")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 600 -P pk -o 2010sheaves.ps 2010sheaves.dvi && ps2pdf 2010sheaves.ps 2010sheaves.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o tmp.ps tmp.dvi")
% (find-pspage  "~/LATEX/tmp.ps")
% (ee-cp "~/LATEX/2010sheaves.pdf" (ee-twupfile "LATEX/2010sheaves.pdf") 'over)
% (ee-cp "~/LATEX/2010sheaves.pdf" (ee-twusfile "LATEX/2010sheaves.pdf") 'over)
% (find-twusfile     "LATEX/" "2010sheaves")
% http://angg.twu.net/LATEX/2010sheaves.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 2010sheaves.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")


\def\colim{\mathop{\mathstrut\rm colim}\limits}
\def\colimxU{\colim_{U \to x}}
\def\colimxU{\colim_{x \ot U}}
\def\UxU{\bigcup\limits_{xÝX}}
\def\UxU{\bigcup\limits_{X \ni x}}
\def\Ucolim{\UxU\!\colimxU}
\def\suto#1{\{s:U \to #1\}}
\def\Top{¦{Top}}

\def\slice#1#2{\begin{pmatrix} #1 \\ \dnto \\ #2 \end{pmatrix}}

On the top square: the adjunction $\LL \dashv \GG$;

On the lower $|·|$: the counit $\ee_Y$ (etalification),

and the unit $\eta_P$ (sheafification).


%D diagram ??
%D 2Dx     100   +75
%D 2D  100 LA    A
%D 2D
%D 2D  +50 B     RB
%D 2D
%D 2D  +35 LRB   A'
%D 2D
%D 2D  +50 B'    RLA
%D 2D
%D 2D  +25 CatB  CatA
%D (( LA .tex= \slice{\Ucolim{P(U)}}{X}  A  .tex= (U^\op\mapsto"P(U))
%D    B  .tex= \slice{Y}{X}              RB .tex= (U^\op\mapsto\suto{Y})
%D    LA A <-| .plabel= a \Lambda
%D    LA B ->  A RB ->
%D    B RB |-> .plabel= b \GG
%D    @ 0 @ 3 harrownodes nil 20 nil <->
%D ))
%D (( LRB .tex= \slice{\Ucolim{\suto{Y}}}{X}
%D    B'  .tex= \slice{Y}{X}
%D    @ 0 @ 1 -> .plabel= l \ee_Y
%D ))
%D (( A'  .tex= (U^\op\mapsto"P(U))
%D    RLA .tex= (U^\op\mapsto\suto{\Ucolim{P(U)}})
%D    @ 0 @ 1 -> .plabel= r \eta_P
%D ))
%D (( CatB .tex= \Top/X
%D    CatA .tex= \Set^{\Opens(X)^\op}
%D    @ 0 @ 1 <- sl^ .plabel= a \Lambda
%D    @ 0 @ 1 -> sl_ .plabel= b \GG
%D ))
%D enddiagram
%D
$$\diag{??}$$


% (find-books "__cats/__cats.el" "moerdijk")
% (find-maclanemoerdijkpage (+ -39   89) "II.6 Sheaves as Étale Spaces")

\def\bbC{{\mathbb{C}}}
\def\bbD{{\mathbb{D}}}
\def\bbE{{\mathbb{E}}}
\def\bbF{{\mathbb{F}}}
\def\bbG{{\mathbb{G}}}
\def\bbH{{\mathbb{H}}}
\def\bbK{{\mathbb{K}}}
\def\bbV{{\mathbb{V}}}
\def\bbW{{\mathbb{W}}}

If $X = \bbV = \dagVee***$ then this gives an adjunction

between $\Top/\bbV$ and $\Set^{\Opens(\bbV)^\op} = \Set^\bbK$.

I know how the sheafification $\eta$ acts on the evil presheaf,

but not yet how $\LL$, $\GG$, $\ee$ work.

%*

\end{document}

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