Bachelor thesis about the Dold-Kan correspondence
https://github.com/Jaxan/Dold-Kan
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87 lines
1.8 KiB
87 lines
1.8 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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\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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$$ \cdots \to C_4 \to^{\del_3} C_3 \to^{\del_2} C_2 \to^{\del_1} C_1 \to^{\del_0} C_0 $$
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zodat $\del_n \circ \del_{n+1} = 0$ voor alle $n \geq 1$.
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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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Bekijk $\Delta^n \to X$, dwz...
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\end{frame}
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\begin{frame}
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\frametitle{Is $\Ch{\cat{Ab}}$ interessant?}
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Gegeven een ketencomplex $C$:
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$$ \cdots \to C_4 \to^{\del_3} C_3 \to^{\del_2} C_2 \to^{\del_1} C_1 \to^{\del_0} C_0 $$
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met $\del_n \circ \del_{n+1} = 0$
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\bigskip
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Dan geldt $im(\del_{n+1}) \trianglelefteq ker(\del_n)$
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Definieer: $H_n(C) = ker(\del_n) / im(\del_{n+1})$
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\end{frame}
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\begin{frame}
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\frametitle{Voorbeeld}
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$ \cdots \to C_1 \to^{\del_0} C_0 $, wat is $H_1 = ker(\del_0) / im(\del_1)$?
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\begin{enumerate}
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\item Triviaal
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\item Niet triviaal
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\end{enumerate}
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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 $N : \cat{sAb} \to \Ch{\cat{Ab}}$.
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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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