@ -12,7 +12,7 @@ There are two special kinds of maps in $\DELTA$, the so called \emph{face} and \
$$\delta_i: [n]\to[n+1], k \mapsto\begin{cases} k &\text{if } k < i;\\ k+1&\text{if } k \geq i. \end{cases}\hspace{0.5cm}0\leq i \leq n+1, \text{ and}$$
$$\sigma_i: [n+1]\to[n], k \mapsto\begin{cases} k &\text{if } k \leq i;\\ k-1&\text{if } k > i. \end{cases}\hspace{0.5cm}0\leq i \leq n$$
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. So in a certain sense, these are all the maps we need to consider. We can now picture the category $\DELTA$ as follows.
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 follows.
\begin{figure}[h!]
\label{fig:delta_cat}
@ -20,9 +20,12 @@ for each $n \in \N$. The nice things about these maps is that every map in $\DEL
\caption{The category $\DELTA$ with the face and degeneracy maps.}
\end{figure}
\todo{sAb: Epi-mono factorization of $\DELTA$}
\begin{definition}
An simplicial abelian group $A$ is a contravariant functor:
$$A: \DELTA^{op}\to\Ab.$$
\end{definition}
Now the category $\sAb$ is defined as the category $\Ab^{\DELTA^{op}}$. 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 $F: \DELTA^{op}\to Ab$. 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.
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}
\label{fig:simplicial_abelian_group}
@ -30,6 +33,8 @@ Now the category $\sAb$ is defined as the category $\Ab^{\DELTA^{op}}$. Because
\caption{A simplicial abelian group.}
\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 $\Delta^{n+1}$. \todo{sAb: add pictures, along the lines of Friedman} 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 $\sigma_i$?}
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?}
@ -47,12 +52,32 @@ Of course the maps $\delta_i$ and $\sigma_i$ in $\DELTA$ satisfy certain equatio
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 a explicit definition of simplicial abelien groups. In this definition a simplicial abelian group $A$ consists of a family abelian groups $(A_n)_{n}$ together with face and degeneracy maps (which are grouphomomorphisms) such that the simplicial equations hold.
\todo{sAb: Write this out, and define notation for it (eg. $\delta^i, \sigma^i$)}
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?}.
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.
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}
$\Delta[n]$
\end{definition}
\todo{sAb: as example do $\Delta[n]$}
\todo{sAb: as example do the free abelian group pointwise}
Joshua Moerman
12 years ago