bsc-thesis/thesis/DoldKan.tex

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

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\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.

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