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\title{Dold-Kan Correspondence}
\author{Joshua Moerman}
\date{\today}

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\textsc{Radboud University Nijmegen}
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Bachelor Thesis in Mathematics
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\emph{Author:}\\
Joshua Moerman\\
3009408
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\emph{Supervisor:} \\
Moritz Groth
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\section*{Contents}
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\section*{Introduction}
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
$$ K: \Ch{\Ab} \simeq \sAb :N, $$
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:
$$ \pi_n(A) \iso H_n(N(A)) \text{ for all } n \in \N. $$

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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