@ -224,7 +224,7 @@ As we are interested in simplicial abelian groups, it would be nice to make thes
\subsection{The Yoneda lemma}
\subsection{The Yoneda lemma}
Recall that the Yoneda lemma stated: $\mathbf{Nat}(y(C), F)\iso F(C)$, where $F:\cat{C}^{op}\to\Set$ is a functor and $C$ an object. In our case we consider functors $X: \DELTA^{op}\to\Set$ and objects $[n]$. So this gives us the natural bijection:
Recall that the Yoneda lemma stated: $\mathbf{Nat}(y(C), F)\iso F(C)$, where $F:\cat{C}^{op}\to\Set$ is a functor and $C$ an object. In our case we consider functors $X: \DELTA^{op}\to\Set$ and objects $[n]$. So this gives us the natural bijection:
$$ X_n \iso\Hom{\sSet}{\Delta[n]}{X}. $$
$$ X_n \iso\Hom{\sSet}{\Delta[n]}{X}. $$
So we can regard $n$-simplices in $X$ as maps from $\Delta[n]$ to $X$. This also extends to the abelian case, where we get an natural isomorphism (of abelian groups):
So we can regard $n$-simplices in $X$ as maps from $\Delta[n]$ to $X$. This also extends to the abelian case, where we get a natural isomorphism (of abelian groups):
\begin{lemma}\emph{(The abelian Yoneda lemma)}
\begin{lemma}\emph{(The abelian Yoneda lemma)}
Let $A$ be a simplicial abelian group. Then there is a group isomorphism
Let $A$ be a simplicial abelian group. Then there is a group isomorphism
In this thesis we will look at a correspondence which was discovered by A. Dold \cite{dold} and D. Kan \cite{kan} independently, hence it is called the \emph{Dold-Kan correspondence}. Abstractly it is the following equivalence of categories:
In this thesis we will look at a correspondence which was discovered by A. Dold \cite{dold} and D. Kan \cite{kan} independently, hence it is called the \emph{Dold-Kan correspondence}. Abstractly it is the following equivalence of categories: