There are generally two definitions of a \emph{simplicial abelian group}, an abstract one and a very explicit one. We will start with the abstract one, luckily it can still be visualised in pictures, then we will derive the explicit definition.
$$\delta_i: [n]\to[n+1], k \mapsto\begin{cases} k &\text{if } k < i;\\ k+1&\text{if } k \geq i. \end{cases}\hspace{0.5cm}0\leq i \leq n+1, \text{ and}$$
for each $n \in\N$. The nice things about these maps is that every map in $\DELTA$ can be decomposed to a composition of these maps. \todo{sAb: Epi-mono factorization of $\DELTA$} So in a sense, these are all the maps we need to consider. We can now picture the category $\DELTA$ as in figure~\ref{fig:delta_cat}.
Althoug this is a very abstract definition, a more geometric intuition can be given. In $\DELTA$ we can regard $[n]$ as an abstract version of the $n$-simplex $\Delta^n$. The maps face maps $\delta_i$ are then exactly maps which point out how we can embed $\Delta^n$ in $\Delta^{n+1}$. This is shown in figure~\ref{fig:delta_cat_geom}. This picutre shows the images of the face maps, for example the image of $\delta_3$ from $[2]$ to $[3]$ is the set $\{0,1,2\}$, which is the bottom face of the tetrahedron. The degeneracy maps are harder to visualize, one can think of them as collapsing maps, where two points are identified with eachother. \todo{sAb: how to draw $\sigma_i$?}
\begin{figure}
\includegraphics{delta_cat_geom}
\caption{The category $\DELTA$ with the face maps shown in a geometric way.}
\label{fig:delta_cat_geom}
\end{figure}
This category $\DELTA$ will act as a protoype for these kind of geometric structures in other categories. This leads to the following definition.
So the category of all simplicial abelian groups, $\sAb$, is the functor category $\Ab^{\DELTA^{op}}$, where morphisms are natural transformations. Because the face and degeneracy maps give all the maps in $\DELTA$ it is sufficient to define images of $\delta_i$ and $\sigma_i$ in order to define a functor $A: \DELTA^{op}\to Ab$. And hence we can picture a simplicial abelian group as done in figure~\ref{fig:simplicial_abelian_group}. Comparing this to figure~\ref{fig:delta_cat} we see that the arrows are reversed, because $A$ is a contravariant functor.
Because a simplicial abelien group $A$ is a contravariant functor, these equations (which only consist of compositions and identities) also hold in its image. For example the first equation would look like: $ A(\delta_i)A(\delta_j)= A(\delta_{j-1})A(\delta_i)$ for $ i < j$ (again note that $A$ is contravariant, and hence composition is reversed). This can be used for an explicit definition of simplicial abelien groups. In this definition a simplicial abelian group $A$ consists of a collection abelian groups $(A_n)_{n}$ together with face and degeneracy maps (which are grouphomomorphisms) such that the simplicial equations hold. More precisely:
\emph{(Explicitly)} An simplicial abelian group $A$ consists of a collection abelian groups $A_n$ together with face maps $d_i : A_n \to A_{n-1}$ and degenracy maps $s_i : A_n \to A_{n+1}$ for $0\leq i \leq n$ and $n \in\N$, such that:
It is already indicated that a functor from $\DELTA^{op}$ to $\Ab$ is determined when the images for the face and degeneracy maps in $\DELTA$ are provided. So gives this a way of restoring the first definition from this one. Conversely, we can apply functorialty to obtain the second definition from the first. So these definitions are the same \todo{sAb: is it ok not to prove this?}. So from now on we will denote $A([n])$ by $A_n$, $A(\sigma_i)$ by $s_i$ and $A(\delta_i)$ by $d_i$, whenever we have a simplicial abelien group $A$.
When using a simplicial abelian group to construct another object, it is often handy to use this second definition, as it gives you a very concrete objects to work with. On the other hand, constructing this might be hard (as you would need to provide a lot of details), in this case we will often use the more abstract definition.
Of course the abstract definition of simplicial abelian group can easilty be generalized to other categories. For example $\Set^{\DELTA^{op}}=\sSet$ is the category of simplicial sets. There are very important simplicial sets:
Note that indeed $\Hom{\DELTA}{X}{[n]}\in\Set$, because the collection of morphisms in a category is per definition a set. We do not need to specify the face or degeneracy maps, as we already know that $\mathbf{Hom}$ is a functor (in both arguments). Still it is useful to write out some cases.
\begin{example}
We will compute how $\Delta[0]$ look like. Note that $[0]$ is an one-element set, so for any set $X$, there is only one function $\ast : X \to[0]$. Hence $\Delta[0]_n =\{\ast\}$ for all $n$. The face and degeneracy maps are now functions from $\{\ast\}$ to $\{\ast\}$. Again there is only one, namely $\id : \{\ast\}\to\{\ast\}$. This gives:
$\Delta[1]$ is a bit more interesting, but still not too hard. We will compute the first three abelian groups $\Delta[1]_0$, $\Delta[1]_1$ and $\Delta[1]_2$. We can use the fact that any monotone increasing map $f: [n]\to[m]$ is a composition of first applying degeneracy maps, and then face maps, ie.: $f: [n]\tot{\sigma_{i_0}\cdots\sigma_{i_M}}[k]\tot{\delta_{j_0}\cdots\delta_{j_N}}[m]$, where $k \leq m, n$.
For $\Delta[1]_0$ we have to consider maps from $[0]$ to $[1]$, we cannot first apply degeneracy maps (there is no object $[-1]$). So this leaves us with the face maps: $\Delta[1]_0=\{\delta_0, \delta_1\}$. For $\Delta[1]_1$ we of course have the identity function and two functions $\delta_0\sigma_0, \delta_1\sigma_0$. Now $\Delta[1]_2$ are the maps from $[2]$ to $[1]$.
We will compute the two face maps $d_0$ and $d_1$ from $\Delta[1]_1$ to $\Delta[1]_0$. Recall that the $\mathbf{Hom}$-functor in the first argument (the contravariant argument) works with precomposition. So this gives:
Where we in the first calculation used the identity law. In the second and third line we used the third simplicial equation, asserting that $\sigma_0\delta_0=\id$. Similarly we can calculate the face map $d_1$:
As we are interested in simplicial abelian group, it would be nice to make these $n$-simplices into simplicial abelian groups. We have seen how to make an abelian group out of any set using the free abelian group. We can use this functor $\Z[-] : \Set\to\Ab$ to induce a functor $\Z^\ast[-] : \sSet\to\sAb$ as shown in the diagram~\ref{fig:diagram_Z}.
\begin{figure}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes]{
\DELTA^{op}&\Set\\
&\Ab\\
};
\path[->]
(m-1-1) edge node[auto] {$ X $} (m-1-2)
(m-1-2) edge node[auto] {$\Z[-]$} (m-2-2)
(m-1-1) edge node[auto] {$ X' $} (m-2-2);
\end{tikzpicture}
\caption{The simplicial set $X$ can be made into a simplicial abelian group $X'$ by postcomposing with $\Z[-]$}
We can apply this to the standard $n$-simplex $\Delta[1]$. This gives $\Delta[1]_0\iso\Z^2$, since $\Delta[1]_0$ had two elements, and $\Delta[1]_1\iso\Z^3$, where the isomorphisms are taken such that:
