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\documentclass[14pt]{beamer}
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% beamer definieert 'definition' al, maar dan engels :(
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% fix van:
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% http://tex.stackexchange.com/questions/38392/how-to-rename-theorem-or-lemma-in-beamer-to-another-language
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\usepackage[dutch]{babel}
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\uselanguage{dutch}
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\languagepath{dutch}
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\deftranslation[to=dutch]{Definition}{Definitie}
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\deftranslation[to=dutch]{Example}{Voorbeeld}
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\definecolor{todocolor}{rgb}{1, 0.3, 0.2}
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\newcommand{\td}[1]{\colorbox{todocolor}{*\footnote{TODO: #1}}}
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\newcommand{\from}{\leftarrow}
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\usepackage{array}
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\input{../thesis/preamble}
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\graphicspath{ {../presentation2/images/} {../thesis/images/} }
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\title{De Dold-Kan correspondentie
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\huge $$ \Ch{\Ab} \simeq \sAb $$}
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\author{Joshua Moerman}
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\institute[Radboud Universiteit Nijmegen]{Begeleid door Moritz Groth}
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\date{}
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\begin{document}
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\begin{frame}
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\titlepage
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\end{frame}
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\begin{frame}{Categorie\"en}
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Een \emph{categorie} $\cat{C}$ bestaat uit
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\begin{center}
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\includegraphics{cat_th}
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\end{center}
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met \emph{compositie} $-\circ-$, zodat
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\begin{itemize}
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\item er is een \emph{identiteit} $\id_c: C \to C$ en
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\item compositie is associatief.
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Voorbeelden}
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\begin{itemize}
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\item[$\Set$]
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objecten: verzamelingen \\
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pijlen: functies
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\item[$\Ab$]
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objecten: abelse groepen \\
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pijlen: groupshomomorfismes
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\item[$\cat{\underline{4}}$]
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\tikz[baseline=-0.5ex]{
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\matrix (m) [matrix of math nodes, row sep=2em, column sep=2em, ampersand replacement=\&]{
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\ast_1 \& \ast_2 \\
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\ast_3 \& \ast_4 \\
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};
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\path[->] (m-1-1) edge node[font=\small, auto] {$ a $} (m-1-2);
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\path[->] (m-1-1) edge node[font=\small, auto] {$ f $} (m-2-1);
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\path[->] (m-1-2) edge node[font=\small, auto] {$ b $} (m-2-2);
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\path[->] (m-2-1) edge node[font=\small, auto] {$ g $} (m-2-2);
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} \hspace{1cm} met $ba = gf$.
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Functors}
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Een \emph{functor} $F: \cat{C} \to \cat{D}$ is een functie op objecten \'en pijlen.
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\begin{center}
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\includegraphics[scale=0.9]{cat_functor}
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\end{center}
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Zodat
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\begin{itemize}
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\item $F(\id_C) = \id_{F(C)}$ en
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\item $F(g \circ f) = F(g) \circ F(f)$.
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Voorbeeld functor}
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Voor een verzameling $V$ definieer
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$$ \Z[V] = \{ \phi: V \to \Z \I \phi(v) \neq 0 \text{ voor eindig veel } v \}. $$
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\bigskip
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Voor een functie $f: V \to W$ definieer
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\begin{gather*}
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\Z[f]: \Z[V] \to \Z[W] \\
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\Z[f](\phi) = \sum_v \phi(v) \chi_{\{f(v)\}}.
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\end{gather*}
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\bigskip
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Dit is een functor: $\Z[-]: \Set \to \Ab$.
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\end{frame}
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\begin{frame}
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\frametitle{Voorbeeld functor}
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Definieer $F: \cat{\underline{4}} \to \Ab$ als volgt:
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$$ F(\ast_1) = F(\ast_2) = F(\ast_3) = F(\ast_4) = \Z $$
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en op pijlen:
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\begin{align*}
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F(f)(n) = 4n & & F(g)(n) = 3n \\
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F(a)(n) = 6n & & F(b)(n) = 2n.
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\end{align*}
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\begin{columns}
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\begin{column}{0.5\textwidth}
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\tikz[baseline=-0.5ex]{
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\matrix (m) [matrix of math nodes, row sep=2em, column sep=2em, ampersand replacement=\&]{
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\Z \& \Z \\
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\Z \& \Z \\
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};
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\path[->] (m-1-1) edge node[font=\small, auto] {$ \times 6 $} (m-1-2);
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\path[->] (m-1-1) edge node[font=\small, auto] {$ \times 4 $} (m-2-1);
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\path[->] (m-1-2) edge node[font=\small, auto] {$ \times 2 $} (m-2-2);
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\path[->] (m-2-1) edge node[font=\small, auto] {$ \times 3 $} (m-2-2);
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}
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\end{column}
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\begin{column}{0.5\textwidth}
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Compositie is behouden, want het diagram commuteert.
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}
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\frametitle{Samenvattend}
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\begin{itemize}
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\item Categorie $\stackrel{D}{=}$ objecten + pijlen.
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\item Functor $\stackrel{D}{=}$ pijl tussen categorie\"en.
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\end{itemize}
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\begin{itemize}
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\item Functor $\sim$ Constructies.
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\item Functor $\sim$ Diagrammen.
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\end{itemize}
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\bigskip\pause
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$F$ is \emph{contravariant} (notatie $F: \cat{C}^{op} \to \cat{D}$) als
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\begin{columns}
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\begin{column}{0.7\textwidth}\includegraphics[scale=0.8]{cat_contrafunctor}\end{column}
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\begin{column}{0.3\textwidth}\small $F(g \circ f) = F(g) \circ F(f)$.\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}
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\frametitle{Belangrijke categorie in mijn scriptie}
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\begin{itemize} \item[$\DELTA$]
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objecten: $[n] = \{0, \ldots, n\}$, $n\in\N$ \\
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pijlen: monotoon stijgende functies.
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\end{itemize}
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\bigskip
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\only<1>{\begin{example}
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Voor elke $n \in \N$ zijn er pijlen
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\end{example}}
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\only<2->{\begin{lemma}
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Elke pijl in $\DELTA$ is een compositie van
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\end{lemma}}
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\begin{itemize}
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\item $\delta_i: [n] \mono [n+1]$ slaat $i$ over \hfill ($0 \leq i \leq n$)
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\item $\sigma_i: [n+1] \epi [n]$ bereik $i$ twee keer \hfill ($0 \leq i < n$)
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\end{itemize}
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\visible<3>{
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Dus $\DELTA = \vcenter{\hbox{\includegraphics{delta_cat}}}$
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}
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\end{frame}
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\begin{frame}
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\frametitle{Belangrijke categorie in mijn scriptie}
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$\DELTA = \vcenter{\hbox{\includegraphics[scale=0.8]{delta_cat_geom}}}$
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\pause\bigskip
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\begin{lemma}
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De \emph{cosimpliciale vergelijkingen} gelden:
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\small
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\begin{align*}
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\delta_j\delta_i &= \delta_i\delta_{j-1}, \hspace{1.5cm} \textnormal{ if } i < j,\\
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\sigma_j\delta_i &= \delta_i\sigma_{j-1}, \hspace{1.5cm} \textnormal{ if } i < j,\\
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\sigma_j\delta_j &= \sigma_j\delta_{j+1} = \id,\\
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\sigma_j\delta_i &= \delta_{i-1}\sigma_j, \hspace{1.5cm} \textnormal{ if } i > j+1,\\
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\sigma_j\sigma_i &= \sigma_i\sigma_{j+1}, \hspace{1.5cm} \textnormal{ if } i \leq j.
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\end{align*}
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\end{lemma}
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\end{frame}
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\begin{frame}
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\begin{center}
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\Large \visible<2->{$A:$} $\DELTA^{op} \to \Ab$ \visible<2->{\hspace{1cm}}
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\bigskip
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\visible<2->{
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$$ A :=
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\begin{tikzpicture}[baseline=-0.5ex]
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\matrix (m) [matrix of math nodes, ampersand replacement=\&, row sep=2em, column sep=2em] {
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A_0 \& A_1 \& A_2 \& \cdots \\
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};
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\draw [raise line=-5, <-] (m-1-1) -> node[font=\small, above] {$ A(\delta_0) $} (m-1-2);
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\draw [raise line=5, <-] (m-1-1) -> node[font=\small, below] {$ A(\delta_1) $} (m-1-2);
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\foreach \r in {0} \draw [raise line=\r, ->] (m-1-1) -> (m-1-2);
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\foreach \r in {-10, 0, 10} \draw [raise line=\r, <-] (m-1-2) -> (m-1-3);
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\foreach \r in {-5, 5} \draw [raise line=\r, ->] (m-1-2) -> (m-1-3);
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\foreach \r in {-15, -5, 5, 15} \draw [raise line=\r, <-] (m-1-3) -> (m-1-4);
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\foreach \r in {-10, 0, 10} \draw [raise line=\r, ->] (m-1-3) -> (m-1-4);
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\end{tikzpicture}$$
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}
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\end{center}
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\end{frame}
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\begin{frame}
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\frametitle{De categorie $\sAb$}
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\begin{itemize}
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\item[Objecten] \emph{Simpliciaal abelse groepen} $A$ \\
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preciezer: functoren $A: \DELTA^{op} \to \Ab$
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\item[Pijlen] \emph{Natuurlijke transformaties} \\
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preciezer: $\phi: A \to B$ bestaat uit $\phi_n: A_n \to B_n$ zodat \\
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\tikz[baseline=-0.5ex]{
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\matrix (m) [matrix of math nodes, row sep=2em, column sep=2em, ampersand replacement=\&]{
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A_n \& A_m \\
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B_n \& B_m \\
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};
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\path[->] (m-1-1) edge node[font=\small, auto] {$ A(f) $} (m-1-2);
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\path[->] (m-1-1) edge node[font=\small, auto] {$ \phi_n $} (m-2-1);
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\path[->] (m-1-2) edge node[font=\small, auto] {$ \phi_m $} (m-2-2);
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\path[->] (m-2-1) edge node[font=\small, auto] {$ B(f) $} (m-2-2);
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} \hspace{1cm} voor alle $f:[m] \to [n]$.
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{De categorie $\Ch{\Ab}$}
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\begin{itemize}
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\item[Objecten] \emph{Ketencomplexen} $C$ \\
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preciezer: collectie abelse groepen $C_n$ en groepshomonorfismes $\del_{n+1}: C_{n+1} \to C_n$ zodat $\del \circ \del = 0$
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\item[Pijlen] \emph{Ketenafbeeldingen} \\
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preciezer: $\phi: C \to D$ bestaat uit $\phi_n: C_n \to D_n$ zodat \\
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\tikz[baseline=-0.5ex]{
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\matrix (m) [matrix of math nodes, row sep=2em, column sep=2em, ampersand replacement=\&]{
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C_{n+1} \& C_n \\
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D_{n+1} \& D_n \\
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};
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\path[->] (m-1-1) edge node[font=\small, auto] {$ \del $} (m-1-2);
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\path[->] (m-1-1) edge node[font=\small, auto] {$ \phi_{n+1} $} (m-2-1);
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\path[->] (m-1-2) edge node[font=\small, auto] {$ \phi_n $} (m-2-2);
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\path[->] (m-2-1) edge node[font=\small, auto] {$ \del $} (m-2-2);
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}
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\end{itemize}
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\end{frame}
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\begin{frame}{$\sAb$ lijkt op $\Ch{\Ab}$}
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Simpliciaal abelse groepen:
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\begin{center}
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\includegraphics{simplicial_abgrp} \\
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met de 5 vergelijkingen
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\end{center}
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Ketencomplexen:
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\begin{center}
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$ C_0 \from C_1 \from C_2 \from \cdots $ \\
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met $\del \circ \del = 0$
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\end{center}
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\pause $\sAb$ heeft meer structuur?
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\end{frame}
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\begin{frame}{De Dold-Kan correspondentie}
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$$ \visible<2->{N:} \sAb \only<1>{\simeq} \only<2->{\rightleftarrows} \Ch{\Ab} \visible<2->{:K} $$
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\visible<2->{
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\begin{align*}
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\text{Zodat}\qquad &\forall C \in \Ch{\Ab}: &N(K(C)) \iso C \\
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\text{en}\qquad &\forall A \in \sAb: &K(N(A)) \iso A.
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\end{align*}
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}
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\bigskip\visible<3->{
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$N$ is in zekere zin surjectief: $\forall C \in \Ch{\Ab}$ is er een $A \in \sAb$ met $N(A) \iso C$.
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}
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\end{frame}
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\begin{frame}{Eerste gok}
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Definieer $M: \sAb \to \Ch{\Ab}$ met $M(A)_n = A_n$.
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\bigskip\pause
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Zij $C = \Z \from 0 \from 0 \from \cdots$\\
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Is er een $A$ zodat $M(A) \iso C$?\\
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M.a.w. $A_0 \iso \Z$ en $A_1 \iso 0$, kan dat?
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\bigskip\pause
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Nee! Want $A_0 \tot{A(\sigma_0)} A_1$ is injectief!\\
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(want $\sigma_0 \delta_0 = \id$, dus $A(\delta_0)A(\sigma_0) = \id$)
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\end{frame}
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\begin{frame}{Definities}
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Zij $A \in \sAb$ \\
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$x \in A_n$ heet een \emph{$n$-simplex} \\
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$x \in A_n$ is \emph{gedegenereerd} als $x = A(\sigma_i)(y)$ voor een zekere $i$ en $y$.
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\end{frame}
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\begin{frame}{De juiste constructie}
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Zij $A \in \sAb$, definieer
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\begin{align*}
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N(A)_n &= \bigcap_{i=1}^n \ker(A(\delta_i)) \\
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\del &= A(\delta_0)
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\end{align*}
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\pause
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\begin{lemma}
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$x \in N(A)_n$ is niet-gedegenereerd.
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\end{lemma}
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\bigskip
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\begin{lemma}
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\centering$ A_n = N(A)_n \oplus D_n(A). $
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\end{lemma}
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\end{frame}
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\begin{frame}{Voorbeeld}
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Definieer de volgende simpliciaal abelse groep:
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\begin{gather*}
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A_n = \Z \\
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A(\delta_i) = A(\sigma_i) = \id.
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\end{gather*}
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\pause
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$$ N(A) = \Z \from 0 \from 0 \from \cdots. $$
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\end{frame}
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\begin{frame}
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\begin{center}
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$$ N: \sAb \rightleftarrows \Ch{\Ab} :K $$
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\pause\bigskip
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\Huge Vragen?
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\end{center}
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\end{frame}
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\end{document}
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