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.
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: $\Hom{\DELTA}{[n]}{[p]}=\{ f : [n]\to[p]\I f(i)\leq f(j)\text{ for all } i < j \}$.
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 \emph{hits $i$ twice}. More precisely it is defined for all $n \in\N$ as:
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.
This is called the epi-mono factorization, because it factors any map $\eta$ into a surjective part ($\sigma_{j_b}\cdots\sigma_{j_1}$) and an injective part ($\delta_{i_a}\cdots\delta_{i_1}$). In a diagram:
{\centering\begin{tikzpicture}
\matrix (m) [row sep=1em, column sep=3em, matrix of math nodes]{
% Note: [] have a meaning in tikz, so I wrapped them in \left \right
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, suppose also $\eta=\delta_{i'_{a'}}\cdots\delta_{i'_1}\sigma_{j'_{b'}}\cdots\sigma_{j'_1}$ such that $0\leq j'_{b'} < \cdots < j'_1 < m$ and $0\leq i'_1 < \cdots < i'_{a'}\leq n$. It is immediately clear that $b = b'$ must hold by counting the elements which are hit twice, and therefore also $a = a'$. Note that $\eta(j'_k)=\eta(j'_{k+1})$, because the sequences are ordered in the same way, this means $j_k = j'_k$ for all $k$. Similarly $i_k$ = $i'_k$ for all $k$.
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.
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.
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.
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:
\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:
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.
We will proof the existence with induction 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 the epi-mon factorization (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 $y = z$.
Now assume $x =\beta^\ast y =\gamma^\ast y$ with $\gamma, \beta : [n]\epi[m]$ such that $\beta\neq\gamma$, and $y \in X_m$ non-degenerate. Then we can find an $\alpha:[m]\to[n]$ such that $\beta\alpha=\id$ and $\gamma\alpha\neq\id$. With the epi-mono factorization write $\gamma\alpha=\delta_{i_a}\cdots\delta_{i_1}\sigma_{j_b}\cdots\sigma_{j_1}$, then by functoriality of $X$:
$$ y =\alpha^\ast\beta^\ast y =\alpha^\ast\gamma^\ast y = s_{j_1}\cdots s_{j_b} d_{i_1}\cdots d_{i_a} y. $$
Note that $y$ was non-degenerate, so $s_{j_1}\cdots s_{j_b}=\id$, hence $d_{i_1}\cdots d_{i_a}=\id$. So $\gamma\alpha=\id$, which gives a contradiction. So $\beta$ is unique.
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:
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.
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 $d_1$:
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).
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.
Note that the set of natural transformations between two simplicial abelian groups $A$ and $B$ is also an abelian group. The proof that $\sAb$ is a preadditive category is very similar to the proof we saw in section~\ref{sec:Chain Complexes}. For two natural transformations $f,g: A \to B$ we simply define $f+g$ pointwise: $(f+g)_n = f_n + g_n$.
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.
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.
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:
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:
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):
The only thing that we need to check is that this bijection preserves the group structure. Recall that this bijection from $\Hom{\sAb}{\Z^\ast[\Delta[n]]}{A}$ to $A_n$ is given by (where $\id=\id_{[n]}$ is a generator in $\Z^\ast[\Delta[n]]$):
$$\phi(f)= f_n(\id)\in X_n \quad\text{ for } f: \Delta[n]\to X. $$
where we regard $\id\in\Delta[a]$ as en element $\id\in\Z^\ast\Delta[n]$, we can do so by the unit of the adjunction. So this bijection is also a group homomorphism, hence we have an isomorphism $A_n \iso\Hom{\sAb}{\Z^\ast[\Delta[n]]}{A}$ of abelian groups.
