Basic definition and stub DoldKan.tex

thesis/DoldKan.tex

@@ -21,6 +21,24 @@
 \begin{document}
 \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.
+
+
 
 % \listoftodos
 % \nocite{*}

thesis/preamble.tex

@@ -8,6 +8,9 @@
 \newcommand{\N}{\mathbb{N}}
 \newcommand{\Z}{\mathbb{Z}}
 \newcommand{\cat}[1]{\mathbf{#1}}
+\newcommand{\Ch}[1]{\mathbf{Ch}(#1)}
+
+\newcommand{\iso}{\cong}
 \newcommand{\eps}{\varepsilon}
 \newcommand{\I}{\,\mid\,}
 \newcommand{\then}{\Rightarrow}