@ -11,23 +11,36 @@ Before defining \emph{simplicial abelian groups}, we will first discuss the more
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:
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:
$$\sigma_i: [n+1]\to[n], k \mapsto\begin{cases} k &\text{if } k \leq i;\\ k-1&\text{if } k > i. \end{cases}\hspace{0.5cm}0\leq i \leq n. $$
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}
\begin{lemma}\emph{(Epi-mono factorization)}
\label{le:epimono}
Let $\eta : [m]\to[n]$ be a map in $\DELTA$. Then $\eta$ can be uniquely decomposed as:
such that $0\leq j_b < \cdots < j_1 < m$ and $0\leq i_1 < \cdots < i_a \leq n$.
\end{lemma}
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, \todo{sAb: uniqueuness epi-mono}
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 $k$.
\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.
@ -77,7 +90,7 @@ Of course the maps $\delta_i$ and $\sigma_i$ in $\DELTA$ satisfy certain relatio
\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...}
By writing out the definitions given above. \todo{sAb: Is this rude?}
\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:
@ -103,13 +116,16 @@ Note that because of the third equation, the degeracy maps $s_i$ are injective.
\end{definition}
\begin{lemma}
\label{le:non-degenerate}
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.
We can write any $x \in X_n$ uniquely as $x =\beta^\ast y$with$\beta : [n]\epi[m]$ a surjective map 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 $x = s_i \beta^\ast y =(\beta\sigma_i)^\ast y$.
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 $y = z$.
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: $\gamma=\beta$}
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.
\end{proof}
\subsection{The standard $n$-simplex}
@ -232,8 +248,10 @@ So we can regard $n$-simplices in $X$ as maps from $\Delta[n]$ to $X$. This also
which is natural in $A$ and $[n]$.
\end{lemma}
\begin{proof}
By using the (non-abelian) Yoneda lemma and the fact that $\Z^\ast$ is a left-adjoint, we already have a natural bijection. The only thing that we need to check is that this bijections preserves the group structure. Recall that the bijection from the (non-abelian) Yoneda lemma is given by:
$$\phi(f)= f_n(\id)\in X_n \text{ for } f: \Delta[n]\to X. $$
By using the (non-abelian) Yoneda lemma and the fact that $\Z^\ast$ is a left-adjoint, we already have a natural bijection:
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 $\Z^\ast[\Delta[n]]$):
$$\phi(f)= f_n(\id)\in X_n \quad\text{ for } f: \Delta[n]\to X. $$
Now let $A$ be a simplicial abelian group and $f, g: \Z^\ast\Delta[n]\to A$ maps. Then we compute:
Comparing chain complexes and simplicial abelian groups, we see a similar structure. Both objects consists of a sequence of abelian groups, with maps in between. At first sight simplicial abelian groups have more structure, because there are maps in both directions. It is not clear how to make degeneracy maps given a chain complex, in fact it is already unclear how to define more maps (the face maps) out of one (the boundary one). Constructing a chain complex from a simplicial abelian group on the other hand seems doable.
Comparing chain complexes and simplicial abelian groups, we see a similar structure. Both objects consists of a sequence of abelian groups, with maps in between. At first sight simplicial abelian groups have more structure, because there are maps in both directions. It is not clear how to make degeneracy maps given a chain complex, in fact it is already unclear how to define more maps (the face maps) out of one (the boundary map). Constructing a chain complex from a simplicial abelian group on the other hand seems doable.
\subsection{Unnormalized chain complex}
Given a simplicial abelian group $A$, we have a family of abelian groups $A_n$. For every $n>0$ we define a group homomorphism $\del_n : A_n \to A_{n-1}$:
Joshua Moerman
12 years ago