Joshua Moerman
12 years ago
1 changed files with 59 additions and 19 deletions
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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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\input{../thesis/preamble} |
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\title{Dold-Kan correspondentie} |
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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{Dold-Kan Correspondentie} |
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\huge $$ \cat{Ch(Ab)} \simeq \cat{sAb} $$ |
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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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\section{Ketencomplex} |
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\begin{frame} |
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\frametitle{Ketencomplex} |
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\begin{definition} |
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Een \emph{ketencomplex} $C$ bestaat uit abelse groepen $C_n$ en homomorfismes $\del_n : C_{n+1} \to C_n$, zodat $\del_n \circ \del_{n+1} = 0$ voor alle $n \in \N$. |
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\end{definition} |
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\pause |
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\bigskip |
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Met andere woorden: |
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$$ \cdots \to C_4 \to C_3 \to C_2 \to C_1 \to C_0 $$ |
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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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Uit $\del_n \circ \del_{n+1} = 0$ volgt $im(\del_{n+1}) \trianglelefteq ker(\del_n)$ |
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\pause |
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Definieer: $H_n(C) = ker(\del_n) / im(\del_{n+1})$ |
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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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\begin{center} |
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\Huge Vragen? |
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\end{center} |
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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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