Bachelor thesis about the Dold-Kan correspondence
https://github.com/Jaxan/Dold-Kan
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47 lines
1.9 KiB
47 lines
1.9 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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$$ \Ch{\cat{Ab}} \simeq \cat{sAb} $$
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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 $N: \cat{sAb} \to \Ch{\cat{Ab}}$ is one half of the equivalence.
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\section{Chain Complexes}
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\begin{definition}
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A chain complex $C$ is a collection of abelian groups $C_n$ together with boundary operators $\del_n: C_{n+1} \to C_n$, such that $\del_n \circ \del_{n+1} = 0$. The collections of all such objects will be denoted by $\Ch{\cat{Ab}}$.
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\end{definition}
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In other words a chain complex is the following diagram.
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$$ \cdots \to C_4 \to C_3 \to C_2 \to C_1 \to C_0 $$
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Of course we can make this more general by taking for example $R$-modules instead of abelian groups. We will later see which kind of algebraic objects make sense to use in this definition.
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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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