Bachelor thesis about the Dold-Kan correspondence https://github.com/Jaxan/Dold-Kan
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\documentclass[14pt]{beamer}
% beamer definieert 'definition' al, maar dan engels :(
% fix van:
% http://tex.stackexchange.com/questions/38392/how-to-rename-theorem-or-lemma-in-beamer-to-another-language
\usepackage[dutch]{babel}
\uselanguage{dutch}
\languagepath{dutch}
\deftranslation[to=dutch]{Definition}{Definitie}
\usepackage{array}
\input{../thesis/preamble}
\title{Dold-Kan correspondentie
\huge $$ \Ch{\cat{Ab}} \simeq \cat{sAb} $$}
\author{Joshua Moerman}
\institute[Radboud Universiteit Nijmegen]{Begeleid door Moritz Groth}
\date{}
\begin{document}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}
\frametitle{Wat is $\Ch{\cat{Ab}}$?}
\begin{definition}
Een \emph{ketencomplex} $C$ bestaat uit abelse groepen met groepshomomorfisme:
$$ \cdots \to C_4 \tot{\del_3} C_3 \tot{\del_2} C_2 \tot{\del_1} C_1 \tot{\del_0} C_0 $$
zodat $\del_n \circ \del_{n+1} = 0$ voor alle $n \in \N$.
\end{definition}
\end{frame}
\begin{frame}
\frametitle{Voorbeeld}
\centering \vspace{-0.5cm}
Bekijk $\Delta^n \tot{f} X$,\, dwz.\, \raisebox{-.2\height}{\includegraphics{simplex_in_X}}
\bigskip
\bigskip
\includegraphics<1>{singular_chaincomplex1}
\includegraphics<2>{singular_chaincomplex2}
\includegraphics<3>{singular_chaincomplex3}
\end{frame}
\begin{frame}
\frametitle{Interessant?}
Gegeven een ketencomplex $C$: \\
$ \cdots \tot{\del_2} C_2 \tot{\del_1} C_1 \tot{\del_0} C_0 $ met $\del_n \circ \del_{n+1} = 0$
\bigskip\bigskip
Dan geldt $im(\del_{n+1}) \trianglelefteq ker(\del_n)$
Definieer: $H_n(C) = ker(\del_{n-1}) / im(\del_n)$
met $ker(\del_{-1}) = C_0$
\end{frame}
\begin{frame}
\frametitle{Voorbeeld}
\raisebox{-.2\height}{\includegraphics[width=0.7\textwidth]{singular_chaincomplex_small}}, $ H_1 = \frac{ker(\del_0)}{im(\del_1)} $?
\bigskip
\begin{tabular}{m{0.3\textwidth} m{0.7\textwidth}}
\includegraphics<1>{singular_homology1}
\includegraphics<2->{singular_homology2}
&
$\sigma_1 - \sigma_2 + \sigma_3 \in ker (\del_0) $ \newline
\visible<2->{$\del_1(\tau) = \sigma_1 - \sigma_2 + \sigma_3 $ \newline
Dus $ \sigma_1 + \sigma_2 - \sigma_3 \in im (\del_1) $ \newline
Dus $ 0 = [\sigma_1 - \sigma_2 + \sigma_3] \in H_1 $}
\end{tabular}
\bigskip
\visible<3->{
\begin{tabular}{m{0.3\textwidth} m{0.7\textwidth}}
\includegraphics{singular_homology3}
&
$ \sigma_1 - \sigma_2 + \sigma_3 \in ker (\del_0) $ \newline
Maar $ \sigma_1 - \sigma_2 + \sigma_3 \not \in im (\del_1) $ \newline
Dus $ 0 \neq [\sigma_1 - \sigma_2 + \sigma_3] \in H_1 $
\end{tabular}
}
\end{frame}
\begin{frame}
\frametitle{Dold-Kan Correspondentie}
\begin{center}
{\Large $ \Ch{\cat{Ab}} \simeq \cat{sAb} $}
verder:
{\Large $$ H_n(N(X)) \iso \pi_n(X) $$}
waarbij $N : \cat{sAb} \tot{\simeq} \Ch{\cat{Ab}}$.
\end{center}
\end{frame}
\begin{frame}
\begin{center}
\Huge Vragen?
\end{center}
\end{frame}
\end{document}