Bachelor thesis about the Dold-Kan correspondence
https://github.com/Jaxan/Dold-Kan
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102 lines
3.4 KiB
102 lines
3.4 KiB
\documentclass[titlepage, 11pt]{amsproc}
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% Activate to begin paragraphs with an empty line rather than an indent
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% toc/refs clickable
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\usepackage{hyperref}
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\input{../thesis/preamble}
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\graphicspath{ {../thesis/images/} }
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\usepackage{wallpaper}
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\title{Dold-Kan Correspondence}
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\author{Joshua Moerman}
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\date{\today}
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\begin{document}
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\begin{titlepage}
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\ThisCenterWallPaper{1.0}{bgtitlepage}
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\centering
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\vspace{10cm}
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\includegraphics[scale=0.2]{ru}\\
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\textsc{Radboud University Nijmegen}
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\vspace{3cm}
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{\huge \bfseries \makeatletter\@title\makeatother}\\
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\vspace{0.3cm}
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Bachelor Thesis in Mathematics
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\vspace{3cm}
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\begin{minipage}{0.4\textwidth}
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\begin{flushleft} \large
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\emph{Author:}\\
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Joshua Moerman\\
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3009408
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\end{flushleft}
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\end{minipage}
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\begin{minipage}{0.4\textwidth}
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\begin{flushright} \large
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\emph{Supervisor:} \\
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Moritz Groth
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\end{flushright}
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\end{minipage}
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\vfill
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\makeatletter\@date\makeatother
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\end{titlepage}
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\section*{Contents}
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\renewcommand\contentsname{}
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\tableofcontents
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\section*{Introduction}
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In this thesis we will study the Dold-Kan correspondence, a celebrated result which belongs to the field of homological algebra or simplicial homotopy theory. Abstractly, one version of the theorem states that there is an equivalence of categories
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$$ K: \Ch{\Ab} \simeq \sAb :N, $$
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where $\Ch{\Ab}$ is the category of chain complexes and $\sAb$ is the category of simplicial abelian groups. This theorem was discovered by A.~Dold \cite{dold} and D.~Kan \cite{kan} independently in 1957. Objects of either of these categories have important invariants. A more refined statement of this equivalence tells us that there is an natural isomorphism between homology groups of chain complexes and homotopy groups of simplicial abelian groups. A bit more precise:
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$$ \pi_n(A) \iso H_n(N(A)) \text{ for all } n \in \N. $$
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In the first section some definitions from category theory are recalled, which are especially important in Sections~\ref{sec:Simplicial Abelian Groups} and \ref{sec:Constructions}. In Section~\ref{sec:Chain Complexes} we will discuss the category of chain complexes and in the end of this section a motivation from algebraic topology will be given for these objects. Section~\ref{sec:Simplicial Abelian Groups} then continues with the second category involved, $\sAb$. This section start with a slightly more general notion and it will be illustrated to have a geometrical meaning. In Section~\ref{sec:Constructions} the correspondence will be defined and proven. In the last section (Section~\ref{sec:Homotopy}) the refined statement will be proven and in the end some more general notes about topology and homotopy will be given, justifying once more the beauty of this correspondence.
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\newpage
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\input{../thesis/1_CategoryTheory}
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\newpage
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\input{../thesis/2_ChainComplexes}
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\newpage
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\input{../thesis/3_SimplicialAbelianGroups}
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\newpage
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\input{../thesis/4_Constructions}
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\newpage
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\input{../thesis/5_Homotopy}
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\newpage
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\listoftodos
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\nocite{*}
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\bibliographystyle{alpha}
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\bibliography{references}
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\end{document}
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