Bachelor thesis about the Dold-Kan correspondence
https://github.com/Jaxan/Dold-Kan
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42 lines
1.4 KiB
42 lines
1.4 KiB
\documentclass[12pt]{amsproc}
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% a la fullpage
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\usepackage{geometry}
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\geometry{a4paper}
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\geometry{twoside=false}
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% Activate to begin paragraphs with an empty line rather than an indent
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\usepackage[parfill]{parskip}
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\setlength{\marginparwidth}{2cm}
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\newtheorem{theorem}{Theorem}[section]
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\newtheorem{definition}[theorem]{Definition}
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\newtheorem{lemma}[theorem]{Lemma}
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\input{../thesis/preamble}
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\title{Dold-Kan Correspondence}
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\author{Joshua Moerman}
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\begin{document}
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\maketitle
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\section*{Introduction}
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In this thesis we will look at a correspondence which was discovered by A. Dold and D. Kan independently, hence it is called the \emph{Dold-Kan correspondence}. Abstractly it is the following equivalence of categories:
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$$ \Ch{\Ab} \simeq \sAb $$
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It is interesting because objects on the left hand side are considered to be algebraic of nature, whereas objects on the right are more topological. In particular this correspondence also gives a isomorphism between homology groups (on the left hand side) and homotopy groups (on the right hand side). 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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where $N: \sAb \to \Ch{\Ab}$ is one half of the equivalence.
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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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\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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