Bachelor thesis about the Dold-Kan correspondence
https://github.com/Jaxan/Dold-Kan
You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
117 lines
2.9 KiB
117 lines
2.9 KiB
\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}
|
|
\graphicspath{ {../presentation/images/} }
|
|
|
|
\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}
|
|
|