Bachelor thesis about the Dold-Kan correspondence
https://github.com/Jaxan/Dold-Kan
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116 lines
2.8 KiB
116 lines
2.8 KiB
\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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\usepackage{array}
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\input{../thesis/preamble}
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\title{Dold-Kan correspondentie
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\huge $$ \Ch{\cat{Ab}} \simeq \cat{sAb} $$}
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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}
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\frametitle{Wat is $\Ch{\cat{Ab}}$?}
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\begin{definition}
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Een \emph{ketencomplex} $C$ bestaat uit abelse groepen met groepshomomorfisme:
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$$ \cdots \to C_4 \tot{\del_3} C_3 \tot{\del_2} C_2 \tot{\del_1} C_1 \tot{\del_0} C_0 $$
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zodat $\del_n \circ \del_{n+1} = 0$ voor alle $n \in \N$.
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\end{definition}
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\end{frame}
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\begin{frame}
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\frametitle{Voorbeeld}
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\centering \vspace{-0.5cm}
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Bekijk $\Delta^n \tot{f} X$,\, dwz.\, \raisebox{-.2\height}{\includegraphics{simplex_in_X}}
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\bigskip
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\bigskip
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\includegraphics<1>{singular_chaincomplex1}
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\includegraphics<2>{singular_chaincomplex2}
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\includegraphics<3>{singular_chaincomplex3}
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\end{frame}
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\begin{frame}
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\frametitle{Interessant?}
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Gegeven een ketencomplex $C$: \\
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$ \cdots \tot{\del_2} C_2 \tot{\del_1} C_1 \tot{\del_0} C_0 $ met $\del_n \circ \del_{n+1} = 0$
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\bigskip\bigskip
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Dan geldt $im(\del_{n+1}) \trianglelefteq ker(\del_n)$
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Definieer: $H_n(C) = ker(\del_{n-1}) / im(\del_n)$
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met $ker(\del_{-1}) = C_0$
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\end{frame}
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\begin{frame}
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\frametitle{Voorbeeld}
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\raisebox{-.2\height}{\includegraphics[width=0.7\textwidth]{singular_chaincomplex_small}}, $ H_1 = \frac{ker(\del_0)}{im(\del_1)} $?
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\bigskip
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\begin{tabular}{m{0.3\textwidth} m{0.7\textwidth}}
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\includegraphics<1>{singular_homology1}
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\includegraphics<2->{singular_homology2}
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&
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$\sigma_1 - \sigma_2 + \sigma_3 \in ker (\del_0) $ \newline
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\visible<2->{$\del_1(\tau) = \sigma_1 - \sigma_2 + \sigma_3 $ \newline
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Dus $ \sigma_1 + \sigma_2 - \sigma_3 \in im (\del_1) $ \newline
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Dus $ 0 = [\sigma_1 - \sigma_2 + \sigma_3] \in H_1 $}
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\end{tabular}
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\bigskip
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\visible<3->{
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\begin{tabular}{m{0.3\textwidth} m{0.7\textwidth}}
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\includegraphics{singular_homology3}
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&
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$ \sigma_1 - \sigma_2 + \sigma_3 \in ker (\del_0) $ \newline
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Maar $ \sigma_1 - \sigma_2 + \sigma_3 \not \in im (\del_1) $ \newline
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Dus $ 0 \neq [\sigma_1 - \sigma_2 + \sigma_3] \in H_1 $
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\end{tabular}
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}
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\end{frame}
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\begin{frame}
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\frametitle{Dold-Kan Correspondentie}
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\begin{center}
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{\Large $ \Ch{\cat{Ab}} \simeq \cat{sAb} $}
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verder:
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{\Large $$ H_n(N(X)) \iso \pi_n(X) $$}
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waarbij $N : \cat{sAb} \tot{\simeq} \Ch{\cat{Ab}}$.
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\end{center}
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\end{frame}
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\begin{frame}
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\begin{center}
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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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