bsc-thesis/thesis/DoldKan.tex

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\title{Dold-Kan Correspondence}
\author{Joshua Moerman}
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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 a 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 other category involved, the category of simplicial abelian groups. This section starts 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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\section*{Conclusion}
In this thesis we have seen two interesting mathematical structures. On one hand there are simplicial sets and simplicial abelian groups which are defined in a abstract and categorical way. The definition was quite short and elegant, nevertheless the objects have a very rich geometrical structure. On the other hand there are chain complexes which have a very simple definition, which are at first sight completely algebraic.

A proof was given of the equivalence of these structures. In this proof we had to take a close look at degenerated simplices in simplicial abelian groups. Some abstract machinery from category theory, like the Yoneda lemma, allowed us to easily construct the needed isomorphisms.

The category of simplicial sets is the abstract framework for doing homotopy theory. Using free abelian groups allowed us to linearize this, resulting in the category of simplicial abelian groups. The Dold-Kan correspondence assures us that there is no loss of information when passing to chain complexes. This makes the category of chain complexes and homological algebra very suitable for doing homotopy theory in a linearized fashion.

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