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Warning: this is an htmlized version!
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% (setenv "TEXINPUTS" "~/LATEX:")
% (setenv "TEXINPUTS" nil)
% (defun c () (interactive) (find-LATEXsh "run-latex 'stem=2014sfc-slides;pla'"))
% (defun c () (interactive) (find-LATEXsh "run-latex 'stem=2014sfc-slides;d4;pla'"))
% (defun d () (interactive) (find-xpdfpage "~/LATEX/2014sfc-slides.pdf"))
% (defun fra () (interactive) (insert "\n% <>\n%\n\\begin{frame}\n\\frametitle{}\n\\end{frame}\n"))
% http://angg.twu.net/LATEX/2014sfc-slides.pdf
% (find-pen-links)
% (find-sh0 "cp -v 2014sfc-slides.pdf /tmp/pen/ebl2014-ochs.pdf")
% «.title-page» (to "title-page")
% «.where-sheaves-stand» (to "where-sheaves-stand")
% «.CT-why» (to "CT-why")
% «.CT-why-now» (to "CT-why-now")
% «.CT-should-be» (to "CT-should-be")
% «.CT-lambda-calculus» (to "CT-lambda-calculus")
% «.ZSets» (to "ZSets")
% «.ZDAGs» (to "ZDAGs")
% «.truth-values» (to "truth-values")
% «.intuitionistic-truth-values» (to "intuitionistic-truth-values")
% «.the-order-topology» (to "the-order-topology")
% «.priming» (to "priming")
% «.unpriming» (to "unpriming")
% «.priming-theorems» (to "priming-theorems")
% «.glueing-functions» (to "glueing-functions")
% «.sheafness» (to "sheafness")
% «.topological-sheafness» (to "topological-sheafness")
% «.the-evil-presheaf» (to "the-evil-presheaf")
% «.dual-operations» (to "dual-operations")
% «.where-are-the-theorems» (to "where-are-the-theorems")
%
% «.closure-and-ess» (to "closure-and-ess")
% (find-LATEX "README")
% (find-dn4file ".files.sh" "cptexinputs")
% (find-sh "pwd; ~/dednat4/.files.sh -n rmtexinputs")
% (find-sh "pwd; ~/dednat4/.files.sh -n cptexinputs")
% (find-sh "pwd; ~/dednat4/.files.sh rmtexinputs")
% (find-sh "pwd; ~/dednat4/.files.sh cptexinputs")
% (find-BEAMERfile "foo-ornate.tex")
% (find-BEAMERfile "foo.tex")
% (find-node "(kpathsea)Index" "TEXINPUTS")
% (find-es "tex" "TEXINPUTS")
% (find-LATEXfile "" "edrx08.sty")
\documentclass[hyperref=colorlinks]{beamer}
% \usepackage[colorlinks]{hyperref} % (find-es "tex" "hyperref")
\usepackage{etex} % (find-es "tex" "no-room-for-a-new-dimen")
\mode<presentation>
{
% \usetheme{Warsaw}
\usetheme{Boadilla}
\useinnertheme{default}
\useinnertheme{rectangles}
\setbeamercovered{transparent}
% or whatever (possibly just delete it)
}
\usepackage[english]{babel}
\usepackage[latin1]{inputenc}
\usepackage{times}
\usepackage[T1]{fontenc}
\usepackage{edrx08}
\input 2011ebl-defs.tex
% (find-LATEX "2011ebl-abs.tex" "priming")
%L process "edrx08.sty" -- (find-dn4ex "edrx08.sty")
\input 2014sfc-slides.dnt
\def\BPM{\mathsf{BPM}}
\def\bbK{\mathbb{K}}
\def\dn{{\dnto}}
\def\Cont{\mathcal{C}}
\def\Cinfty{\mathcal{C}^‚}
\def\ctor#1{\mathcal{C}^‚(#1,\R)}
\def\ctos#1{C(#1)}
\def\bigdagop{
\def\searrow{\nwarrow}
\def\swarrow{\nearrow}
\def\downarrow{\uparrow}
}
\def\bigdagnoarrows{
\def\searrow{}
\def\swarrow{}
\def\downarrow{}
}
%
\begin{document}
% (find-beamerugpage 73 "\\setbeamertemplate{navigation symbols}{}")
% (find-beamerugtext 73 "\\setbeamertemplate{navigation symbols}{}")
\setbeamertemplate{navigation symbols}{}
% «title-page» (to ".title-page")
%
\title{Sheaves for Children}
\author{Eduardo Ochs}
\institute[PURO/UFF]{
Departmento de Física e Matemática\\
Pólo Universitário de Rio das Ostras\\
UFF\\
% \footnotesize
\scriptsize
\url{http://angg.twu.net/math-b.html} \\
\url{http://angg.twu.net/ferramentas-para-ativistas.html} \\
\tiny \phantom{a}
{\color{gray}(Version: 2014apr09)} \\
}
\date{EBL 2014}
\begin{frame}
\titlepage
\end{frame}
% «where-sheaves-stand» (to ".where-sheaves-stand")
%
\begin{frame}[fragile]
\frametitle{Where sheaves stand}
{\footnotesize
\begin{verbatim}
____________________
| |
| Category Theory | Cartesian Closed Categories,
| ________________ | Lambda-Calculus,
| | | | Intuitionistic Logic
| | Topos Theory | |
| | ____________ | |
| | | | | | Modal Logic (S4)
| | | Sheaves | | |
| | |____________| | | Topology
| |________________| |
|____________________| Algebraic Geometry
\end{verbatim}
}
\end{frame}
% «CT-why» (to ".CT-why")
%
\begin{frame}[fragile]
\frametitle{CT: Why?}
\par Category Theory is fascinating (for some people!),
\par but (usually) too abstract...
\msk
\par The right level of abstraction
\par makes lots of proofs {\it almost} automatic:
\par {\it proving} something in CT means
\par {\it constructing} something (CT is constructive!), and
\par all ``natural'' constructions are equivalent (``coherence'').
\msk
\par More or less like this:
\begin{quote}
\par Let A and B be (arbitrary) sets.
\par Then there is an ``obvious'' function $\text{flip}: A×B \to B×A$.
\end{quote}
% \msk
\par This {\sl ought} to make some parts of CT {\sl easy}!!!
\msk
\par (Long story... see ``Internal Diagrams and Archetypal Reasoning
in Category Theory'')
\end{frame}
% «CT-why-now» (to ".CT-why-now")
%
\begin{frame}
\frametitle{Why study CT {\sl now}?}
\includegraphics[width=8.5cm]{garota_de_ipanema.jpg}
\par Public education in Brazil is being dismantled -
\par maybe we should be doing better things than studying
\par very technical \& inaccessible subjects
\par with no research grants -
\end{frame}
% «CT-should-be» (to ".CT-should-be")
%
\begin{frame}
\frametitle{Category theory should be more accessible}
\par Most texts about CT are for specialists in research universities...
\par {\sl Category theory should be more accessible.}
\msk
\par To whom?...
\begin{itemize}
\item Non-specialists (in research universities)
\item Grad students (in research universities)
\item Undergrads (in research universities)
\item Non-specialists (in conferences - where we have to be quick)
\item Undergrads (e.g., in CompSci - in teaching colleges) - (``Children'')
\end{itemize}
\end{frame}
% «ZSets» (to ".ZSets")
%
\begin{frame}
\frametitle{ZSets}
\par Take a finite, non-empty subset of $\N^2$;
\par translate it lowerleftwards as most as possible in $\N^2$,
\par until you get something that touches both axes.
\msk
\par Subsets of $\N^2$ obtained in this way are said to be
\par ``well-positioned'', and we call them {\sl ZSets}.
\msk
\par We can use a positional notation with bullets
\par to denote our favourite ZSets (unambiguously!)...
\msk
\begin{tabular}{|l|l|c|l|} \hline
% Lambda & $\dagLambda***$ & \\ \hline
$V$ & Vee & $\dagVee***$ & $\{(0,1),\, (2,1),\, (1,0)\}$ \\ \hline
$K$ & Kite & $\dagKite*****$ & $\{(1,3),\, (0,2),\, (2,2),\, (1,1),\, (1,0)\}$ \\ \hline
$H$ & House & $\dagHouse*****$ & $\{(1,2),\, (0,2),\, (2,2),\, (0,1),\, (2,1)\}$ \\ \hline
\end{tabular}
\msk
\par $\uparrow$ Some of my favorite ZSets -
\par note that they have both short, one-letter names
\par and long, pronounceable names.
\msk
\end{frame}
% «ZDAGs» (to ".ZDAGs")
%
\begin{frame}
\frametitle{ZDAGs}
\par An arrow between points of $\N^2$ that goes one unit down
\par and 0/1/-1 units horizontally is called a {\sl black pawn's move}.
\msk
\par Take a ZSet, $D$, and draw all possible black pawns moves
\par between its points; this gives us a set of arrows, $\BPM(D)$,
\par that turns $D$, a ZSet, into a directed, acyclical, graph, $\bbD$,
\par in a canonical way: $\bbD = (D, \BPM(D)).$
\msk
\par Note the change of font!!!: \quad $D \dashrightarrow \bbD$
%Notation (note the change of font):
% D is a ZSet,
% bbD = (D, BPM(D)) is a _ZDAG_.
\msk
Example:
$$\def\ph{\phantom{m}}
K \;\; = \;\; \dagKite***** \;\; = \quad
{\bigdagnoarrows
\bigKite {(1,3)} {(0,2)\ph} {\ph(2,2)} {(0,1)} {(0,0)}
}
\qquad
\qquad
\bbK \;\; = \quad
\bigKite {(1,3)} {(0,2)\ph} {\ph(2,2)} {(0,1)} {(0,0)}
$$
\end{frame}
% «truth-values» (to ".truth-values")
%
\begin{frame}[fragile]
\frametitle{Truth-values}
\par A function from a ZSet $D$ to $\{0,1\}$ is a {\color{red}\sl
modal truth-value}.
\msk
\par The positional notation gives us a way to write modal
truth-values very compactly, and the points on a ZSet have a natural
order - the ``reading order'', in which we read them line by line,
left to right in each line. \msk \par This gives us a way to {\sl read
aloud} modal truth-values - \par and to list all modal truth-values
in order.
$$\bigKite abcde
\qquad
\bigKite 00110
\;\; = \;\; \dagKite00110 \;\; = \;\; P
\qquad
(\text{``Kite 00110''})
$$
\msk
Notation: $\Pts(\bbD)$ is the set of modal truth-values on $\bbD$
We use ``$\Pts(\bbD)$'' because $\dagKite00110$ ``is'' $\{c,d\}$ ($\subset K$)
\end{frame}
% «intuitionistic-truth-values» (to ".intuitionistic-truth-values")
%
\begin{frame}
\frametitle{Intuitionistic truth-values}
\msk
$$\bigKite a b c d e \qquad \bigKite 0 0 1 1 0 \;\; = P \qquad (\text{``Kite 00110''})
$$
\msk
\par Now consider that each 1 in $P$ is covered with (wet) black paint.
\par Then $P$ (``Kite 00110'') is not {\sl stable}, because the paint
\par of the 1 in position $d$ will flow down into the 0 of position $e$,
\par and paint it black.
\msk
\par {\color{red}\sl Stable} modal truth-values are called
{\color{red}\sl intuitionistic truth-values}.
\msk
\par Notation: $\dn P$ is $P$ after letting the black paint flow down.
\par Example: $\dn \dagKite 00110 = \dagKite 00111$
\end{frame}
% «the-order-topology» (to ".the-order-topology")
%
\begin{frame}
\frametitle{The order topology}
\par
$$\dn \dagKite 00110 = \dagKite 00111$$
\bsk
\par Notation: $\Pts(\bbD)$ is the set of modal truth-values on $\bbD$
\par Notation: $\Opens(\bbD)$ is the set of intuitionistic truth-values on $\bbD$
%\msk
$$\dn: \Pts(\bbD) \to \Opens(\bbD)$$
\par The topology $\Opens(\bbD)$ is the {\sl order topology} -
\par an arrow $\aa \to \bb$ in $\bbD$ means that
\par if an open set contains $\aa$ it has to contain $\bb$ too.
\msk
\par {\footnotesize (Order topologies are Alexandroff.)}
\end{frame}
% «priming» (to ".priming")
%
\begin{frame}[fragile]
\frametitle{Priming}
\par Amazing fact: very often $\Opens(\bbD)$ can be represented as a ZDAG too!
\par $\Opens(\bbD)$ has a natural order:
\par $P \to Q$ means
$P \le Q$,
$P \subseteq Q$,
$P \vdash Q$,
\par and $§ = \dagVee111$ (``Top'') is the terminal of the category...
\msk
\par But if we draw $\Opens(\bbD)^\op$ instead of $\Opens(\bbD)$
\par we can see clearly how $\bbD \ito \Opens(\bbD)^\op$:
\msk
$$\bigVee a b c
\qquad
\bigKite {\dagVee111} {\dagVee101} {\dagVee 011} {\dagVee 001} {\dagVee 000}
\qquad
\bigKite {\dn ab} {\dn a} {\dn b} {\dn c} {\emp}
$$
\msk
\par Note that $\dn ab = \dn \{a,b\} = \dn \dagVee110 = \dagVee111$.
\par Def: $\bbD' = \Opens(\bbD)^\op$.
\par $\bbV' \cong \bbK$ - and, by abuse of language, $\bbV' = \bbK$.
\end{frame}
% «unpriming» (to ".unpriming")
%
\def\G#1#2#3#4#5#6{\dagGuill{#1}{#2}{#3}{#4}{#5}{#6}}
%
\begin{frame}
\frametitle{Unpriming}
\par If $\bbC' = \bbD$ can we recover $\bbC$ from $\bbD$?
\par Better: if $\bbD$ is a ZDAG that is a Heyting algebra
\par can we find a $\bbC \subset \bbD$ such that $\bbC' = \bbD$?
\par Can we use that to determine quickly whether
\par an arbitrary $\bbD$ is a Heyting algebra?
\par Yes, yes, \& yes!
% Look at the elements of $\left(\dagGuill******\right)'$
% that have exactly one arrow leaving them...
$$\left(
\bigGuill abcdef
\right)'
\quad \cong \quad
\hugeGuillPrime
{\G 111111} % 12
{\G 101111} % 11
{\G 011111} % 10
{\G 001111} % 8
{\G 010111} % 9
{\G 001011} % 7
{\G 000111} % 5
{\G 001010} % 6
{\G 000011} % 4
{\G 000010} % 3
{\G 000001} % 2
{\G 000000} % 1
\qquad
\hugeGuillPrime
{\dn ab} % 111111
{\dn a} % 101111
{\dn bc} % 011111
{\dn cd} % 001111
{\dn b} % 010111
{\dn cf} % 001011
{\dn d} % 000111
{\dn c} % 001010
{\dn ef} % 000011
{\dn e} % 000010
{\dn f} % 000001
{\emptyset } % 000000
$$
\end{frame}
% «priming-theorems» (to ".priming-theorems")
%
\begin{frame}
\frametitle{Priming: theorems}
% \par For each ZDAG $\bbD$, let $\catD$ be $\bbD^*$ seen as a category.
% \par We say that $\bbC \subset \bbD$ when there is a
% \par proper inclusion $\catC \ito \catD$.
\par We say that $\bbD$ is {\em 3-thin} when $\dagHthree*** \not\subset \bbD$.
\par We say that $\bbD$ is {\em square-thin} when $\dagVV**** \not\subset \bbD$.
\par We say that $\bbD$ is {\em thin} when it is both 3-thin and square-thin.
\par Fact: if $\bbD$ is 3-thin then $\bbD'$ is a ZDAG.
\par Fact: if $\bbD$ is thin then $\bbD'$ is a thin ZDAG.
\msk
Fact: every topology - whether planar or not - is a
Heyting algebra - i.e., we can interpret $T$, $F$, $\land$, $\lor$, $\to$,
$\neg$ on it, and every $\bbD'$ is a topology...
\msk
{\color{red}
\par Priming gives us LOTS of Heyting algebras,
\par and lots of {\sl planar} Heyting algebras!
}
\bsk
{\sl Topological} sheaves are defined on diagrams like
$\bbD \ito \bbD' \ito \bbD''$.
\end{frame}
% «glueing-functions» (to ".glueing-functions")
% (find-LATEX "2011ebl-abs.tex" "quotient")
%
\begin{frame}
\frametitle{Glueing locally-defined functions}
\par Let $U$ be $(-‚,1)$
\par and $V$ be $(0,‚)$...
\par Consider these open sets in $\R$,
$$\def\ph{\phantom{m}}
{\bigdagop
\bigKite {(-‚,‚)} {(-‚,1)\ph} {\ph(0,‚)} {(0,1)} {\emp}
\qquad
% \quad
\bigKite {UþV} {U} {V} {UÌV} {\emp}
\qquad
\bigKite {X} {U} {V} {W} {\emp}
}
\qquad
\qquad
\bigKite {\ctor X} {\ctor U\ph\ph} {\ph\ph\ctor V} {\ctor W} {\ctor\emp}
\qquad
\qquad
\bigKite {\ctos X} {\ctos U} {\ctos V} {\ctos W} {\ctos\emp}
$$
\par and the sheaf $C$ of $\Cinfty$ functions from them to $\R$.
\msk
\par Upward arrows are {\sl inclusions} (of an open set into another).
\par Downward arrows are {\sl restrictions} (of domains).
\msk
\par Two functions $f_U$ and $f_V$ are {\sl compatible}
\par if their restrictions to $UÌV$ coincide.
\msk
\par Each compatible family $\{f_U, f_V\}$ in $C$ has a unique glueing $f_X$.
\par Generalize that, and you get the definition of {\sl sheafness}.
\end{frame}
% «sheafness» (to ".sheafness")
%
\begin{frame}
\frametitle{Sheafness}
\par A compatible family $\{f_U, f_V\}$ is defined on $\{U, V\} = \dagKite01100$,
\par and it can be extended, using the restriction maps $\rho_{UW}:FU \to FW$ etc,
\par to a downward-closed compatible family $\{f_U, f_V, f_W, f_\emp\}$,
\par defined on $\{U, V, W, \emp\} = \dagKite01111$...
\msk
\par The ``unique glueing'' $f_X$ of $\{f_U, f_V\}$ can be extended
\par to a downward-closed compatible family $\{f_X, f_U, f_V, f_W, f_\emp\}$,
\par defined on $\{X, U, V, W, \emp\} = \dagKite11111$.
\msk
\par The restriction
$$\{f_X, f_U, f_V, f_W, f_\emp\}
\;\; \mton{\rho} \;\;
\{f_U, f_V, f_W, f_\emp\}
$$
\par is trivial to define -
\par {\color{red}sheafness means that maps like these are bijections.}
\end{frame}
% «topological-sheafness» (to ".topological-sheafness")
%
\begin{frame}
\frametitle{Topological sheafness}
\par A closure operator:
$$÷ \{U, V, W, \emp\} = \{X, U, V, W, \emp\}$$
\par it takes the union $U þ V þ W þ \emp = X$
\par and then all subsets of that.
\par It acts on $\bbV''$: \quad $÷: \dagVee***'' \to \dagVee***''$
$$÷ \dagKite01111 = \dagKite11111$$
\par Which elements of $\bbV''$ are stable by $÷$?
\par Only $\dn\{U,V\} \mto \dn \{X\}$ ($\dagKite01111 \mapsto \dagKite11111$)
and $\emp \mto \{\emp\}$ ($\dagKite00000 \mapsto \dagKite00001$)
\par are {\sl not} stable by $÷$.
\par The stable elements of $\bbV''$ are these: $\dagKitePrime1011110$.
\par {\color{red}These diagrams of stable elements are what we need to
\par define sheaves ``in general''.}
\end{frame}
% «the-evil-presheaf» (to ".the-evil-presheaf")
% (find-LATEXfile "2011ebl-abs.tex" "evil-presheaf")
%
\begin{frame}
\frametitle{The evil presheaf}
A presheaf $F$ in $\Set^{\Opens(\bbV)^\op}$
is simply a functor $F: \Opens(\bbV)^\op \to \Set$.
%
% (find-dn4 "experimental.lua" "relphantom")
%L forths[".xtag="] = function ()
%L local dx, tag = getwordasluaexpr(), getword()
%L tex = "\\ph{"..tag.."}"
%L ds[1] = storenode{x=ds[1].x+dx, y=ds[1].y, tex=tex, tag=tag}
%L end
%
%D diagram evil-presheaf
%D 2Dx 100 +20 +20 +30 +20 +20
%D 2D 100 A0 B0
%D 2D / \ / \
%D 2D v v v v
%D 2D +20 A1 A2 B1 B2
%D 2D \ / \ /
%D 2D v v v v
%D 2D +20 A3 B3
%D 2D | |
%D 2D v v
%D 2D +20 A4 B4
%D 2D
%D (( A0 .tex= E(X)
%D A1 .tex= E(U)
%D A2 .tex= E(V)
%D A3 .tex= E(W)
%D A4 .tex= E(\emp)
%D A0 A1 ->
%D A0 A2 ->
%D A1 A3 ->
%D A2 A3 ->
%D A3 A4 ->
%D ))
%D (( B0 .tex= \{e_X,e'_X\} place .xtag= -6 e_X .xtag= 12 e'_X
%D # B1 .tex= \{e_U,e'_U\} place .xtag= -5 e_U .xtag= 10 e'_U
%D B1 .tex= \{e_U\}
%D B2 .tex= \{e_V,e'_V\} place .xtag= -5 e_V .xtag= 11 e'_V
%D B3 .tex= \{e_W\}
%D B4 .tex= \{e_\emp\}
%D # e_X e_U -> e'_X e'_U ->
%D # e_X e_V -> e'_X e_V ->
%D # e_U B3 -> e'_U B3 ->
%D # e_V B3 -> e'_V B3 ->
%D # B3 B4 ->
%D e_X B1 -> e'_X B1 ->
%D e_X e_V -> e'_X e_V ->
%D B1 B3 ->
%D e_V B3 -> e'_V B3 ->
%D B3 B4 ->
%D ))
%D enddiagram
%D
$$
% (find-LATEXgrep "grep -nH -e color *.tex")
% \def\ph#1{{\color{red}#1}}
% \def\ph#1{\phantom{#1}}
\def\ph{\phantom}
\diag{evil-presheaf}
$$
The {\color{red}evil presheaf} $E: \Opens(\bbV)^\op \to \Set$, above,
fails to be in a sheaf in two ways:
the compatible family $\{e_U, e_V\}$ has two different glueings,
the compatible family $\{e_U, e'_V\}$ doesn't have a glueing.
\end{frame}
% «dual-operations» (to ".dual-operations")
%
\begin{frame}
\frametitle{Dual operations}
\par Due to we being in a finite / planar / etc case,
\par several interesting operations have duals (adjoints):
\msk
\begin{itemize}
\item In finite DAGs the transitive-reflexive closure $(A,R) \mto (A,R^*)$
\par has an adjoint that keeps only the ``essential arrows'' of the graph;
\item The ``let the paint flow down'' operation $\dn \dagKite00110=\dagKite00111$
\par has an adjoint $\dagKite 00111 \mto \dagKite 00100$ that returns
the ``generators''of an open set;
\item Each closure operator like $\calU \mto ÷\calU$ has an adjoint that
returns the smallest equivalent cover...
\end{itemize}
%\par In finite DAGs the transitive-reflexive closure $(A,R) \mto (A,R^*)$
%\par has an adjoint that keeps only the ``essential arrows'' of the graph;
%\msk
%\par The ``let the paint flow down'' operation $\dn \dagKite00110=\dagKite00111$
%\par has an adjoint $\dagKite 00111 \mto \dagKite 00100$ that returns
% the ``generators''of an open set;
%\msk
%\par Each closure operator like $\calU \mto ÷\calU$ has an adjoint that
% returns the smallest equivalent cover...
\end{frame}
% «where-are-the-theorems» (to ".where-are-the-theorems")
%
\begin{frame}
\frametitle{Where are the theorems?}
\par {\color{red}Not here! Why???}
\par Because this is ``for children'' -
\par we are focusing on the tools to let people check particular cases...
\par and this {\sl complements} [Bell 1988] and my IDARCT paper,
\par that explains how to do theorems and archetypal cases in parallel
\msk
\par Slightly more advanced things:
\begin{itemize}
\item CCCs and Heyting Algebras; $(∧Q) \dashv (Q{\to})$
\item presheaves of the form $\Set^\bbD$ as toposes
\item the classifier object of a $\Set^\bbD$
\item other modalities (besides $÷$) in a $\Set^\bbD$
\item all logical properties of modalities follow from three axioms
\item operations on the lattice of modalities
\item forcing
\item sheafification
\item geometric morphisms between toposes - I need help here ${=}($
\end{itemize}
\end{frame}
%
%
%\section*{Summary}
%
%\begin{frame}{Summary}
%\end{frame}
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%% All of the following is optional and typically not needed.
%\appendix
%\section<presentation>*{\appendixname}
%\subsection<presentation>*{For Further Reading}
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% \frametitle<presentation>{For Further Reading}
\frametitle{For Further Reading}
\begin{thebibliography}{10}
\beamertemplatebookbibitems
% Start with overview books.
\bibitem{Bell1988}
J.L.~Bell.
\newblock {\em Toposes and Local Set Theories}.
\newblock Oxford, 1988 (re-ed: Dover, 2007).
\beamertemplatearticlebibitems
% Followed by interesting articles. Keep the list short.
\bibitem{Ochs2013}
E.~Ochs.
\newblock{\em Internal Diagrams and Archetypal Reasoning in Category Theory}
\newblock Logica Universalis, 2013
\newblock{\url{http://angg.twu.net/math-b.html}}.
% (find-TH "math-b" "sheaves-on-zdags")
% (find-TH "math-b" "internal-diags-in-ct")
\end{thebibliography}
\end{frame}
\end{document}
% «closure-and-ess» (to ".closure-and-ess")
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\begin{frame}
\frametitle{Closure and essentiality}
\par If $(A,R)$ is a graph,
\par then $(A,R^*)$ is its transitive-reflexive closure.
\msk
\par Notation and terminology:
\par $\bbD = (D, \BPM(D))$
\par $\bbD^* = (D, \BPM(D)^*)$
\par $\catD$ is $\bbD^*$ seen as a category ($\catD$ a poset)
\par $\Set^\catD$ is a ZDAG-topos
\msk
\par $\Opens(\bbD)$ are opens sets of the order topology on $\bbD$
\par $\bbD' = \Opens(\bbD)$
\msk
\par We ${\color{red}\heartsuit}$ transitive and reflexive relations
\par (and categories, and posets).
\msk
\par The operation $R \mapsto R^*$ is well-known.
\msk
\par If $(A,R)$ is a finite DAG,
\par look for the smallest set of arrows, $R^-$,
\par such that $(R^-)^* = R^*$.
\par Fact 1: on finite DAGs this $R^-$ is unique.
\par Fact 2: for any ZDAG $\bbD$ we have $\bbD^- = (\bbD^*)^- = \bbD$.
\par {\sl This is not well-known.}
% \msk
% \par Example: $\bbK \to \bbK^* \to (\bbK^*)^-$
\end{frame}
%D diagram miniadj
%D 2Dx 100 120
%D 2D 100 a^L <= a
%D 2D - -
%D 2D | |
%D 2D v v
%D 2D 120 b => b^R
%D a^L a <= a^L b |-> a b^R |-> b b^R =>
%D enddiagram
%$$\diag{miniadj}$$
%: [a,b]^1
%: -------
%: [a,b]^1 b b|->c
%: ------- ------------
%: a c
%: ---------------
%: a,c
%: ---------1
%: a,b|->a,c
%:
%: ^tree1
%:
%$$\ded{tree1}$$
% «CT-lambda-calculus» (to ".CT-lambda-calculus")
%
\begin{frame}[fragile]
\frametitle{CT and $\lambda$-calculus}
\msk
\par Again:
\begin{quote}
\par Let A and B be (arbitrary) sets.
\par Then there is an ``obvious'' function $\text{flip}: A×B \to B×A$.
\end{quote}
\par This idea can be formalized in (typed, polymorphic) $\lambda$-calculus -
\par because $\lambda$-calculus is {\sl made} for expressing computations.
\par Roughly, we have this parallel:
\begin{tabular}{|l|l|} \hline
$\lambda$-calculus & Categories \\ \hline
computations & constructions \\
$\lambda$-terms & diagrams \\ \hline
\end{tabular}
Sales pitch:
{\sl Come learn CT, diagrammatic reasoning, $\lambda$-calculus and
functional languages at the same time!}
\end{frame}
% \section{Motivation}
% \subsection{The Basic Problem That We Studied}
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