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

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\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:
$$ \Ch{\cat{Ab}} \simeq \cat{sAb} $$
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 $N: \cat{sAb} \to \Ch{\cat{Ab}}$ is one half of the equivalence.

\section{Chain Complexes}
\begin{definition}
	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 $\Ch{\cat{Ab}}$.
\end{definition}

In other words a chain complex is the following diagram.
$$ \cdots \to C_4 \to C_3 \to C_2 \to C_1 \to C_0 $$

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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