\section{Simplicial Abelian Groups}
\label{sec:Simplicial Abelian Groups}

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.

\subsection{Abstract definition}
\begin{definition}
	We define a category $\DELTA$, where the objects are the finite ordinals $[n] = \{0, \dots, n\}$ and maps are monotone increasing functions.
\end{definition}

There are two special kinds of maps in $\DELTA$, the so called \emph{face} and \emph{degeneracy} maps, defined as (resp.):

$$\delta_i: [n] \to [n+1], k \mapsto \begin{cases} k & \text{if } k < i;\\ k+1 & \text{if } k \geq i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n+1, \text{ and}$$
$$\sigma_i: [n+1] \to [n], k \mapsto \begin{cases} k & \text{if } k \leq i;\\ k-1 & \text{if } k > i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n$$

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}.

\begin{figure}[h!]
	\includegraphics{delta_cat}
	\caption{The category $\DELTA$ with the face and degeneracy maps.}
	\label{fig:delta_cat}
\end{figure}

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.

\begin{definition}
	An simplicial abelian group $A$ is a contravariant functor:
	$$A: \DELTA \to \Ab.$$
	(Or equivalently a covariant functor $A: \DELTA^{op} \to \Ab.$)
\end{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.

\begin{figure}
	\includegraphics{simplicial_abelian_group}
	\caption{A simplicial abelian group.}
	\label{fig:simplicial_abelian_group}
\end{figure}


\subsection{Explicit definition}
Of course the maps $\delta_i$ and $\sigma_i$ in $\DELTA$ satisfy certain equations, these are the so called \emph{simplicial equations}.
\todo{sAb: Is \emph{simplicial equations} really a thing?}

\begin{lemma}
	The face and degeneracy maps in $\DELTA$ satisfy the simplicial equations, ie.:
	\begin{align}
		\delta_j\delta_i &= \delta_i\delta_{j-1}  \hspace{0.5cm} \text{ if } i < j,\\
		\sigma_j\delta_i &= \delta_i\sigma_{j-1}  \hspace{0.5cm} \text{ if } i < j,\\
		\sigma_j\delta_j &= \sigma_j\delta_{j+1} = \id,\\
		\sigma_j\delta_i &= \delta_{i-1}\sigma_j  \hspace{0.5cm} \text{ if } i > j+1,\\
		\sigma_j\sigma_i &= \sigma_i\sigma_{j+1}  \hspace{0.5cm} \text{ if } i \leq j.
	\end{align}
\end{lemma}
\begin{proof}
	By writing out the definitions given above.
\end{proof}

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:

\begin{definition}
	\emph{(Explicitly)} An simplicial abelian group $A$ consists of a collection abelian groups $A_n$ together with face maps $\delta^i : A_n \to A_{n-1}$ and degenracy maps $\sigma^i : A_n \to A_{n+1}$ for $0 \leq i \leq n$ and $n \in \N$, such that:
	\begin{align}
		\delta^i\delta^j &= \delta^{j-1}\delta^i  \hspace{0.5cm} \text{ if } i < j,\\
		\delta^i\sigma^j &= \sigma^{j-1}\delta^i  \hspace{0.5cm} \text{ if } i < j,\\
		\delta^j\sigma^j &= \delta^{j+1}\sigma^j = \id,\\
		\delta^i\sigma^j &= \sigma^j\delta^{i-1}  \hspace{0.5cm} \text{ if } i > j+1,\\
		\sigma^i\sigma^j &= \sigma^{j+1}\sigma^i  \hspace{0.5cm} \text{ if } i \leq j.
	\end{align}
\end{definition}

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 $\sigma^i$ and $A(\delta_i)$ by $\delta^i$, 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.

\todo{sAb: Note that $\sigma^i$ is a monomorphism because of (3)}

\subsection{Other simplicial objects}
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:

\begin{definition}
	The standard $n$-simplex is given by:
	$$\Delta[n] = \Hom{\DELTA}{-}{[n]} : \DELTA^{op} \to \Set.$$
\end{definition}

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).

\todo{sAb: as example do $\Delta[n]$}
\todo{sAb: as example do the free abelian group pointwise}

