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Joshua Moerman 12 years ago
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  1. 2
      thesis/3_SimplicialAbelianGroups.tex
  2. 40
      thesis/DoldKan.tex
  3. BIN
      thesis/images/ru.pdf

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thesis/3_SimplicialAbelianGroups.tex

@ -224,7 +224,7 @@ As we are interested in simplicial abelian groups, it would be nice to make thes
\subsection{The Yoneda lemma} \subsection{The Yoneda lemma}
Recall that the Yoneda lemma stated: $\mathbf{Nat}(y(C), F) \iso F(C)$, where $F:\cat{C}^{op} \to \Set$ is a functor and $C$ an object. In our case we consider functors $X: \DELTA^{op} \to \Set$ and objects $[n]$. So this gives us the natural bijection: Recall that the Yoneda lemma stated: $\mathbf{Nat}(y(C), F) \iso F(C)$, where $F:\cat{C}^{op} \to \Set$ is a functor and $C$ an object. In our case we consider functors $X: \DELTA^{op} \to \Set$ and objects $[n]$. So this gives us the natural bijection:
$$ X_n \iso \Hom{\sSet}{\Delta[n]}{X}. $$ $$ X_n \iso \Hom{\sSet}{\Delta[n]}{X}. $$
So we can regard $n$-simplices in $X$ as maps from $\Delta[n]$ to $X$. This also extends to the abelian case, where we get an natural isomorphism (of abelian groups): So we can regard $n$-simplices in $X$ as maps from $\Delta[n]$ to $X$. This also extends to the abelian case, where we get a natural isomorphism (of abelian groups):
\begin{lemma}\emph{(The abelian Yoneda lemma)} \begin{lemma}\emph{(The abelian Yoneda lemma)}
Let $A$ be a simplicial abelian group. Then there is a group isomorphism Let $A$ be a simplicial abelian group. Then there is a group isomorphism
$$ A_n \iso \Hom{\sAb}{\Z^\ast[\Delta[n]]}{A}, $$ $$ A_n \iso \Hom{\sAb}{\Z^\ast[\Delta[n]]}{A}, $$

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thesis/DoldKan.tex

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\documentclass[11pt]{amsproc} \documentclass[titlepage, 11pt]{amsproc}
% a la fullpage % a la fullpage
\usepackage{geometry} \usepackage{geometry}
@ -9,6 +9,9 @@
\usepackage[parfill]{parskip} \usepackage[parfill]{parskip}
\setlength{\marginparwidth}{2cm} \setlength{\marginparwidth}{2cm}
% toc/refs clickable
\usepackage{hyperref}
\theoremstyle{plain} \theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section] \newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition} \newtheorem{proposition}[theorem]{Proposition}
@ -26,7 +29,40 @@
\author{Joshua Moerman} \author{Joshua Moerman}
\begin{document} \begin{document}
\maketitle \begin{titlepage}
\centering
\vspace{10cm}
\includegraphics[scale=0.2]{ru}\\
\textsc{Radboud University Nijmegen}
\vspace{3cm}
{\huge \bfseries Dold-Kan Correspondence}\\
Bachelor Thesis Mathematics
\vspace{3cm}
\begin{minipage}{0.4\textwidth}
\begin{flushleft} \large
\emph{Author:}\\
Joshua Moerman\\
3009408
\end{flushleft}
\end{minipage}
\begin{minipage}{0.4\textwidth}
\begin{flushright} \large
\emph{Supervisor:} \\
Moritz Groth
\end{flushright}
\end{minipage}
\vfill
\today
\end{titlepage}
\section*{Contents}
\renewcommand\contentsname{}
\tableofcontents
\section*{Introduction} \section*{Introduction}
In this thesis we will look at a correspondence which was discovered by A. Dold \cite{dold} and D. Kan \cite{kan} independently, hence it is called the \emph{Dold-Kan correspondence}. Abstractly it is the following equivalence of categories: In this thesis we will look at a correspondence which was discovered by A. Dold \cite{dold} and D. Kan \cite{kan} independently, hence it is called the \emph{Dold-Kan correspondence}. Abstractly it is the following equivalence of categories:

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thesis/images/ru.pdf

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