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.
64 lines
2.2 KiB
64 lines
2.2 KiB
\documentclass[12pt]{amsproc}
|
|
|
|
% a la fullpage
|
|
\usepackage{geometry}
|
|
\geometry{a4paper}
|
|
\geometry{twoside=false}
|
|
|
|
% Activate to begin paragraphs with an empty line rather than an indent
|
|
\usepackage[parfill]{parskip}
|
|
\setlength{\marginparwidth}{2cm}
|
|
|
|
\theoremstyle{plain}
|
|
\newtheorem{theorem}{Theorem}[section]
|
|
\newtheorem{proposition}[theorem]{Proposition}
|
|
\newtheorem{lemma}[theorem]{Lemma}
|
|
\newtheorem{corollary}[theorem]{Corollary}
|
|
|
|
\theoremstyle{definition}
|
|
\newtheorem{definition}[theorem]{Definition}
|
|
\newtheorem{example}[theorem]{Example}
|
|
\newtheorem{exlemma}[theorem]{Example/Lemma}
|
|
|
|
\input{../thesis/preamble}
|
|
\graphicspath{ {../thesis/images/}, {../presentation/images/} }
|
|
|
|
\title{Dold-Kan Correspondence}
|
|
\author{Joshua Moerman}
|
|
|
|
\begin{document}
|
|
\maketitle
|
|
|
|
\section*{Introduction}
|
|
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:
|
|
$$ \Ch{\Ab} \simeq \sAb $$
|
|
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:
|
|
$$ \pi_n(A) \iso H_n(N(A)) \text{ for all } n \in \N $$
|
|
where $N: \sAb \to \Ch{\Ab}$ is one half of the equivalence.
|
|
|
|
In the first section some definitions from category theory are given, because we will need them later on. Then in the second section we will discuss the first category involved in the correspondence, $\Ch{\Ab}$, the category of chain complexes. The third section then continues with the second category involved, $\sAb$, especially for this section we will need category theory. Then we will look at the coorespondence itself.
|
|
|
|
\newpage
|
|
\input{../thesis/1_CategoryTheory}
|
|
|
|
\newpage
|
|
\input{../thesis/2_ChainComplexes}
|
|
|
|
\newpage
|
|
\input{../thesis/3_SimplicialAbelianGroups}
|
|
|
|
\newpage
|
|
\input{../thesis/4_Constructions}
|
|
|
|
\newpage
|
|
\input{../thesis/5_Homotopy}
|
|
|
|
\newpage
|
|
\todo{References: Lamotke, Friedman, Weibel}
|
|
|
|
\newpage
|
|
\listoftodos
|
|
% \nocite{*}
|
|
% \bibliographystyle{alpha}
|
|
% \bibliography{references}
|
|
\end{document}
|
|
|