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

Before defining \emph{simplicial abelian groups}, we will first discuss the more general notion of \emph{simplicial sets}. There are generally two definitions of simplicial sets, 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. The reader who is interested in how these notions developed, should consider reading the introduction by Friedman \cite{friedman}, which also gives nice illustrations.

\subsection{Abstract definition}
\begin{definition}
	We define a category $\DELTA$, where the objects are the finite ordinals $[n] = \{0, \dots, n\}$ for $n \in \N$ and maps are monotone increasing functions: $\Hom{\DELTA}{[n]}{[p]} = \{ f : [n] \to [p] \I f(i) \leq f(j) \text{ for all } i < j \}$.
\end{definition}

There are two special kinds of maps in $\DELTA$, the so called \emph{face} and \emph{degeneracy} maps. The $i$-th face maps $\delta_i$ is the unique injective monotone increasing function which \emph{omits} $i$. More precisely it is defined for all $n \in \Np$ as (note that we do not explicitly denote $n$ in this notation):
$$ \delta_i: [n-1] \to [n], k \mapsto \begin{cases} k & \text{if } k < i;\\ k+1 & \text{if } k \geq i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n. $$

The $i$-th degeneracy map $\sigma_i$ is the unique surjective monotone increasing function which hits $i$ twice. More precisely it is defined for all $n \in \N$ as:
$$ \sigma_i: [n+1] \to [n], k \mapsto \begin{cases} k & \text{if } k \leq i;\\ k-1 & \text{if } k > i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n. $$

The nice things about these maps is that every map in $\DELTA$ can be decomposed to a composition of these maps. So in a sense, these are all the maps we need to consider.

\begin{lemma}
	\label{le:epimono}
	Let $\eta : [m] \to [n]$ be a map in $\DELTA$. Then $\eta$ can be uniquely decomposed as:
	$$ \eta = \delta_{i_a} \cdots \delta_{i_1} \sigma_{j_b} \cdots \sigma_{j_1}, $$
	such that $0 \leq j_b < \cdots < j_1 < m$ and $0 \leq i_1 < \cdots < i_a \leq n$.
\end{lemma}
\begin{proof}
	We start with the existence. Consider the set $S = \{ k \in [m-1] \I \eta(k) = \eta(k+1) \}$. These are precisely the elements which are hit twice, now let $S = \{ j_1, \ldots, j_{|S|} \}$ with $0 \leq j_{|S|} < \cdots < j_1 < m$. This gives rise to a surjection $\sigma = \sigma_{j_b} \cdots \sigma_{j_1}: [m] \epi [m-|S|]$.

	Similarly consider $T = \{ k \in [m - |S|] \I k \not \in \eta[m] \}$. These are precisely the elements which are omitted, now let $T = \{ i_1, \ldots, i_{|T|} \}$ with $0 \leq i_1 < \cdots < i_{|T|} \leq n$. This gives an injection $\delta = \delta_{i_a} \cdots \delta_{i_1} : [m - |S|] \mono [n]$. Now we see that $\eta = \delta\sigma$.

	Now for uniqueness, \todo{sAb: uniqueuness epi-mono}
\end{proof}

We can now picture the category $\DELTA$ as in figure~\ref{fig:delta_cat}. Note that the face and degeneracy maps are not unrelated. We will make the exact relations precise later.

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

Although 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 face maps $\delta_i$ are then exactly maps which point out how we can embed $[n]$ in $[n+1]$. 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 each other. For example, this collapses a triangle into a line.

\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}
	A \emph{simplicial set} $X$ is a functor:
	$$X: \DELTA^{op} \to \Set.$$
	(Or equivalently a contravariant functor $X: \DELTA \to \Set.$)
\end{definition}

So the category of all simplicial sets, $\sSet$, is the functor category $\Set^{\DELTA^{op}}$, 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 $X: \DELTA^{op} \to \Set$, keep in mind that these should satisfy some relations which we will discuss next. Hence we can picture a simplicial set as done in figure~\ref{fig:simplicial_set}. Comparing this to figure~\ref{fig:delta_cat} we see that the arrows are reversed, because $X$ is a contravariant functor.

\begin{figure}
	\includegraphics{simplicial_set}
	\caption{A simplicial set.}
	\label{fig:simplicial_set}
\end{figure}


\subsection{Explicit definition}
Of course the maps $\delta_i$ and $\sigma_i$ in $\DELTA$ satisfy certain relations, these are the so called \emph{cosimplicial identities}.

\begin{lemma}
	The face and degeneracy maps in $\DELTA$ satisfy the cosimplicial identities, i.e.:
	\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. \todo{sAb: this is a bit rude, maybe write out some of it...}
\end{proof}

Because a simplicial set $X$ 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: $ X(\delta_i)X(\delta_j) = X(\delta_{j-1})X(\delta_i) $ for $ i < j$. This can be used for an explicit definition of simplicial sets. In this definition a simplicial set $X$ consists of a collection sets $X_n$ together with the face and degeneracy maps. More precisely:

\begin{definition}
	\emph{(Explicitly)} An simplicial set $X$ consists of a collection sets $X_n$ together with functions $d_i : X_n \to X_{n-1}$ and $s_i : X_n \to X_{n+1}$ for $0 \leq i \leq n$ and $n \in \N$, such that the simplicial identities hold:
	\begin{align}
		d_i d_j &= d_{j-1} d_i  \hspace{0.5cm} \text{ if } i < j,\\
		d_i s_j &= s_{j-1} d_i  \hspace{0.5cm} \text{ if } i < j,\\
		d_j s_j &= d_{j+1} s_j = \id,\\
		d_i s_j &= s_j d_{i-1}  \hspace{0.5cm} \text{ if } i > j+1,\\
		s_i s_j &= s_{j+1} s_i  \hspace{0.5cm} \text{ if } i \leq j.
	\end{align}
\end{definition}

It is already indicated that a functor from $\DELTA^{op}$ to $\Set$ is determined when the images for the face and degeneracy maps in $\DELTA$ are provided. So this gives 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. From now on we will denote $X([n])$ by $X_n$, $X(\sigma_i)$ by $s_i$ and $X(\delta_i)$ by $d_i$, whenever we have a simplicial set $X$. For any other map $\beta : [n] \to [p]$ we will denote the induced map by $\beta^\ast : X_p \to X_n$.

When using a simplicial set 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.

Note that because of the third equation, the degeracy maps $s_i$ are injective. This means that in the set $X_{n+1}$ there are always ``copies'' of elements of $X_n$. In a way these elements are not interesting, hence we call them degenerate.
\begin{definition}
	An element $x \in X_{n+1}$ is \emph{degenerate} if it lies in the image of $s_i : X_n \to X_{n+1}$ for some $i$. An element is called \emph{non-degenerate} if this is not the case.
\end{definition}
\begin{lemma}
	We can write any $x \in X_n$ uniquely as $x = \beta^\ast y$ for some surjective map $\beta : [n] \epi [m]$ and $y \in X_m$ non-degenerate.
\end{lemma}
\begin{proof}
	We will proof the existence with inducion over $n$. For $n=0$ the statement is trivial, since all elements in $X_0$ are non-degenerate. Assume the statement is proven for $n$. Let $x \in X_{n+1}$. Clearly if $x$ itself is non-degenerate, we can write $x = \id^\ast x$. Otherwise it is of the form $x = s_i x'$ for some $x' \in X_n$ and $i$. The induction hypothesis tells us that we can write $x' = \beta^\ast y$ for some surjection $\beta: [n] \epi [m]$ and $y \in X_m$ non-degenerate. So $x = s_i \beta^\ast y = (\beta \sigma_i)^\ast y$.

	For uniqueness, assume $x = \beta^\ast y = \gamma^\ast z$ with $\beta : [n] \epi [m]$, $\gamma: [n] \epi [m']$ and $y \in X_m, z \in X_{m'}$ non-degenerate. Because $\beta$ is surjective there is an $\alpha:[m]\to[n]$ such that $\beta\alpha = \id$ and hence $y = \alpha^\ast \gamma^\ast z = (\gamma\alpha)^\ast z$. By lemma~\ref{le:epimono} we can write $\gamma\alpha = \delta_{i_a} \cdots \delta_{i_1} \sigma_{j_b} \cdots \sigma_{j_1}$, using that $y$ is non-degenerate we know that $\gamma\alpha$ is injective. So we have $\gamma\alpha: [m] \mono [m']$. Because of symmetry (of $y$ and $z$) we also have some map $[m'] \mono[m]$, so $m = m'$. So $\gamma\alpha$ is also surjective, hence the identity function, thus $y = z$.
	\todo{sAb: $\gamma = \beta$}
\end{proof}

\subsection{The standard $n$-simplex}
Recall that for any category $\cat{C}$ we have the $\mathbf{Hom}$-functor: $\Hom{\cat{C}}{-}{-} : \cat{C}^{op} \times \cat{C} \to \Set$. We can fix an object $C \in \cat{C}$ and get a functor $\Hom{\cat{C}}{-}{C} : \cat{C}^{op} \to \Set$. In our case we can get the following simplicial sets in this way:

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

Note that this is also the definition of the Yoneda embedding $\Delta[n] = y[n]$. In a moment we will see why the Yoneda lemma is useful to us. But we will explicitly describe two of such standaard $n$-simplices.

\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:
	$$ \Delta[0] = \{\ast\} \to \{\ast\} \to \{\ast\} \to \cdots. $$
	Note that the only non-degenerate simplex is the unqiue $0$-simplex.
\end{example}

\begin{example}
	$\Delta[1]$ 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:
	\begin{align*}
		d_0(id) &= \id \delta_0 = \delta_0 \\
		d_0(\delta_0\sigma_0) &= \delta_0 \sigma_0 \delta_0 = \delta_0 \\
		d_0(\delta_1\sigma_0) &= \delta_0 \sigma_0 \delta_0 = \delta_1.
	\end{align*}
	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 $d_1$:
	\begin{align*}
		d_1(id) &= \id \delta_1 = \delta_1 \\
		d_1(\delta_0\sigma_0) &= \delta_0 \sigma_0 \delta_1 = \delta_0 \\
		d_1(\delta_1\sigma_0) &= \delta_0 \sigma_0 \delta_1 = \delta_1.
	\end{align*}

	$$ \Delta[1] =
	\begin{tikzpicture}[baseline=-0.5ex]
	\matrix (m) [matrix of math nodes] { 
		\{\delta_0, \delta_1\} & \{\sigma_0 \delta_0, \id, \sigma_0 \delta_1\} & \{ \} & \cdots \\
	}; 

	\foreach \r in {-5, 5} \draw [raise line=\r, <-] (m-1-1) -> (m-1-2);
	\foreach \r in {0} \draw [raise line=\r, ->] (m-1-1) -> (m-1-2);

	\foreach \r in {-10, 0, 10} \draw [raise line=\r, <-] (m-1-2) -> (m-1-3);
	\foreach \r in {-5, 5} \draw [raise line=\r, ->] (m-1-2) -> (m-1-3);

	\foreach \r in {-15, -5, 5, 15} \draw [raise line=\r, <-] (m-1-3) -> (m-1-4);
	\foreach \r in {-10, 0, 10} \draw [raise line=\r, ->] (m-1-3) -> (m-1-4);
	\end{tikzpicture}.$$
	In this simplicial set there are three non-degenerate simplices. There is $\id \in \Delta[1]_1$, which clearly is non-degenerate, and the two $0$-simplices $\delta_0$ and $\delta_1$. One can think of this simplicial set as a line (the non-degenerate $1$-simplex) with its endpoints (the two $0$-simplices).
\end{example}

\subsection{Other simplicial objects}
Of course the abstract definition of simplicial abelian group can easily be generalized to other categories. For any category $\cat{C}$ we can consider the functor category $\cat{sC} = \cat{C}^{\DELTA^{op}}$. In this thesis we are interested in the category $\sAb = \Ab^{\DELTA^{op}}$ of simplicial abelian groups. So a simplicial abelian group $A$ is a collection of abelian groups $A_n$, together with face and degeneracy maps, which in this case means group homomorphisms $d_i$ and $s_i$ such that the simplicial equations hold.

As we are interested in simplicial abelian groups, it would be nice to make these standard $n$-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 following diagram.
\begin{figure}[h!]
	\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 $\Z[-]$}
	\label{fig:diagram_Z}
\end{figure}
\begin{lemma}
	The functor $\Z^\ast[-] : \sSet \to \sAb$ is a left-adjoint, with $U^\ast: \sAb \to \sSet$ (the pointwise forgetful functor) as right-adjoint.
\end{lemma}
\begin{proof}
	First we note that $U^\ast \Z^\ast [X]_n = U\Z[X_n]$ by definition, so pointwise we get (by the fact that $\Z$ and $U$ already form an adjunction):
\begin{center}
	\begin{tikzpicture}
		\matrix (m) [matrix of math nodes]{
			X_n & U^\ast \Z^\ast[X]_n & \Z[X_n] \\
			    & U(A_n)              & A_n \\
		};
		\path[->]
		(m-1-1) edge node[auto] {$ i $} (m-1-2)
		(m-1-2) edge node[auto] {$ U(\overline{f}) $} (m-2-2)
		(m-1-1) edge node[auto] {$ f $} (m-2-2);
		\path[->]
		(m-1-3) edge node[auto] {$ \overline{f} $} (m-2-3);
	\end{tikzpicture}
\end{center}
	Then use naturality of $i$ (in $X_n$, thus in particular in $n$) to extend this to $i^\ast : X \to U^\ast \Z^\ast [X]$. Now if we're given a natural transformation $f: X \to U^\ast A$ of simplicial sets we can again construct $\overline{f}: \Z^\ast[X] \to A$ pointwise. The reader is invited to check the details.
\end{proof}

\begin{example}
	We can apply this to the standard $n$-simplex $\Delta[1]$. This gives $\Delta[1]_0 \iso \Z^2$, since $\Delta[1]_0$ has two elements, and $\Z^\ast[\Delta[1]]_1 \iso \Z^3$, where the isomorphisms are taken such that:
	\begin{align*}
		\delta_0         &\mapstot{\iso} (1, 0) \\
		\delta_1         &\mapstot{\iso} (0, 1) \\
		\sigma_0\delta_0 &\mapstot{\iso} (1, 0, 0) \\
		\id              &\mapstot{\iso} (0, 1, 0) \\
		\sigma_0\delta_1 &\mapstot{\iso} (0, 0, 1)
	\end{align*}
	The face maps from $\Z^\ast[\Delta[1]]_1$ to $\Z^\ast[\Delta[1]]_0$ under these isomorphisms are then given by:
	\begin{align*}
		d_0(x, y, z) &= (x+y, z) \\
		d_1(x, y, z) &= (x, y+z)
	\end{align*}
\end{example}

\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:
$$ X_n \iso \Hom{\sSet}{\Delta[n], X}. $$
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):
$$ A_n \iso \Hom{\sAb}{\Z^\ast[\Delta[n]], A}, $$
which is natural in $A$ and $[n]$.

\todo{sAb: note use of Yoneda lemma (also abelian)}
