@ -123,7 +123,7 @@ Note that because of the third equation, the degeneracy maps $s_i$ are injective
\begin{proof}
We will proof the existence by 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 $y = z$, meaning that the non-degenerate $m$-simplex is unique.
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 $y =\alpha^\ast\beta^\ast y =\alpha^\ast\gamma^\ast z =(\gamma\alpha)^\ast z$. 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 $y = z$, meaning that the non-degenerate $m$-simplex$y$ is unique.
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 $X$
$$ y =\alpha^\ast\beta^\ast y =\alpha^\ast\gamma^\ast y = s_{j_1}\cdots s_{j_b} d_{i_1}\cdots d_{i_a} y. $$
@ -138,7 +138,7 @@ Recall that for any category $\cat{C}$ we have the $\mathbf{Hom}$-functor $\Hom{
Note that $\Delta[-]: \DELTA\to\sSet$ is exactly the Yoneda embedding. In a moment we will see why the Yoneda lemma is useful to us, but let us first explicitly describe two examples of such standard simplices.
Note that $\Delta[-]: \DELTA\to\sSet$ is exactly the Yoneda embedding. So a $m$-simplex in $\Delta[n]$ is nothing more than a monotone function $[m]\to[n]$. In a moment we will see why the Yoneda lemma is useful to us, but let us first explicitly describe two examples of such standard simplices.
\begin{example}
We will compute how $\Delta[0]$ look like. Note that $[0]$ is an one-element set, so for any set $S$, there is only one function $\ast: S \to[0]$. Hence $\Delta[0]_n =\{\ast\}$ for all $n$ and the face and degeneracy maps are necessarily the identity maps $\id: \{\ast\}\to\{\ast\}$. Thus, $\Delta[0]$ looks like
@ -160,6 +160,8 @@ Note that $\Delta[-]: \DELTA \to \sSet$ is exactly the Yoneda embedding. In a mo
Note that the only non-degenerate simplex is the unique $0$-simplex.
\end{example}
We can characterize the non-degenerate simplices for the other standard $n$-simplices as well. Recall that a simplex is degenerate if it lies the image of $s_i$ for some $i$. In the simplicial set $\Delta[n]$ the degeneracy maps $s_i$ are given by precomposing with $\sigma_i$ (by definition of the $\mathbf{Hom}$-functor). Given a $m$-simplex $x \in\Delta[n]_m$, using the epi-mono factorization we can write it as $x =\delta\sigma: [m]\to[n]$, where $\delta$ is injective and $\sigma$ surjective. It is now easily seen that $x$ is degenerate if and only if $\sigma\neq\id$. In other words a $m$-simplex $x \in\Delta[n]_m$ is non-degenerate if and only if $x: [m]\to[n]$ is injective. Note that for $m>n$ no such injective monotone functions exist and for $m=n$ there is a unique one, namely $\id_{[n]}$.
\begin{example}
$\Delta[1]$ is a bit more interesting, but still not too complicated. We will describe the first three sets $\Delta[1]_0$, $\Delta[1]_1$ and $\Delta[1]_2$. We can use the fact that any monotone function $f: [n]\to[m]$ is a composition of first applying degeneracy maps, and then face maps, i.e.: $f: [n]\tot{\sigma_{i_0}\cdots\sigma_{i_M}}[k]\tot{\delta_{j_0}\cdots\delta_{j_N}}[m]$, where $k \leq m, n$.
@ -223,24 +225,23 @@ This justifies that we may drop this extra decoration ($^\ast$) and write $\Z[-]
\begin{lemma}
The functor $\Z[-]: \sSet\to\sAb$ is a left adjoint, with $U: \sAb\to\sSet$ as right adjoint.
\end{lemma}
\begin{proof}
First we note that $U\Z[X]_n = U\Z[X_n]$ by definition, so pointwise we get (by the fact that $\Z$ and $U$ already form an adjunction):
As this is a purely categorical question (it even works for arbitrary functor categories), only a sketch of the proof is given. First note that by the fact that $\Z$ and $U$ already form an adjunction, and if we are given a natural transformation $f: X \to UA$ of simplicial sets we get the following diagram for each $n \in\N$:
Then use naturality of $i$ (in $X_n$, thus in particular in $n$) to extend this to $i^\ast : X \to U\Z[X]$. Now if we are given a natural transformation $f: X \to UA$ of simplicial sets we can again construct$\overline{f}: \Z[X]\to A$ pointwise. The reader is invited to check the details.
\end{proof}
Then use naturality of $\eta$ (in $X_n$, thus in particular in $n$) to extend this to $\eta : X \to U\Z[X]$. The uniqueness of the maps $\overline{f}_n$ will assure that we get a natural transformation$\overline{f}: \Z[X]\to A$. The reader is invited to check the details.
\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
Joshua Moerman
11 years ago