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.
Joshua Moerman
12 years ago