Definition 14.11.1. Let $U$ be a simplicial set. We say $x$ is an $n$-simplex of $U$ to signify that $x$ is an element of $U_ n$. We say that $y$ is the $j$the face of $x$ to signify that $d^ n_ jx = y$. We say that $z$ is the $j$th degeneracy of $x$ if $z = s^ n_ jx$. A simplex is called degenerate if it is the degeneracy of another simplex.
14.11 Simplicial sets
Let $U$ be a simplicial set. It is a good idea to think of $U_0$ as the $0$-simplices, the set $U_1$ as the $1$-simplices, the set $U_2$ as the $2$-simplices, and so on.
We think of the maps $s^ n_ j : U_ n \to U_{n + 1}$ as the map that associates to an $n$-simplex $A$ the degenerate $(n + 1)$-simplex $B$ whose $(j, j + 1)$-edge is collapsed to the vertex $j$ of $A$. We think of the map $d^ n_ j : U_ n \to U_{n - 1}$ as the map that associates to an $n$-simplex $A$ one of the faces, namely the face that omits the vertex $j$. In this way it become possible to visualize the relations among the maps $s^ n_ j$ and $d^ n_ j$ geometrically.
Here are a few fundamental examples.
Example 14.11.2. For every $n \geq 0$ we denote $\Delta [n]$ the simplicial set We leave it to the reader to verify the following statements. Every $m$-simplex of $\Delta [n]$ with $m > n$ is degenerate. There is a unique nondegenerate $n$-simplex of $\Delta [n]$, namely $\text{id}_{[n]}$.
\[ \Delta ^{opp} \longrightarrow \textit{Sets},\quad [k] \longmapsto \mathop{\mathrm{Mor}}\nolimits _{\Delta }([k], [n]) \]
Lemma 14.11.3. Let $U$ be a simplicial set. Let $n \geq 0$ be an integer. There is a canonical bijection which maps a morphism $\varphi $ to the value of $\varphi $ on the unique nondegenerate $n$-simplex of $\Delta [n]$.
\[ \mathop{\mathrm{Mor}}\nolimits (\Delta [n], U) \longrightarrow U_ n \]
Proof. Omitted. $\square$
Example 14.11.4. Consider the category $\Delta /[n]$ of objects over $[n]$ in $\Delta $, see Categories, Example 4.2.13. There is a functor $p : \Delta /[n] \to \Delta $. The fibre category of $p$ over $[k]$, see Categories, Section 4.35, has as objects the set $\Delta [n]_ k$ of $k$-simplices in $\Delta [n]$, and as morphisms only identities. For every morphism $\varphi : [k] \to [l]$ of $\Delta $, and every object $\psi : [l] \to [n]$ in the fibre category over $[l]$ there is a unique object over $[k]$ with a morphism covering $\varphi $, namely $\psi \circ \varphi : [k] \to [n]$. Thus $\Delta /[n]$ is fibred in sets over $\Delta $. In other words, we may think of $\Delta /[n]$ as a presheaf of sets over $\Delta $. See also, Categories, Example 4.38.7. And this presheaf of sets agrees with the simplicial set $\Delta [n]$. In particular, from Equation (14.4.0.1) and Lemma 14.11.3 above we get the formula for any simplicial set $U$.
\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\Delta )}(\Delta /[n], U) = U_ n \]
Lemma 14.11.5. Let $U$, $V$ be simplicial sets. Let $a, b \geq 0$ be integers. Assume every $n$-simplex of $U$ is degenerate if $n > a$. Assume every $n$-simplex of $V$ is degenerate if $n > b$. Then every $n$-simplex of $U \times V$ is degenerate if $n > a + b$.
Proof. Suppose $n > a + b$. Let $(u, v) \in (U \times V)_ n = U_ n \times V_ n$. By assumption, there exists a $\alpha : [n] \to [a]$ and a $u' \in U_ a$ and a $\beta : [n] \to [b]$ and a $v' \in V_ b$ such that $u = U(\alpha )(u')$ and $v = V(\beta )(v')$. Because $n > a + b$, there exists an $0 \leq i \leq a + b$ such that $\alpha (i) = \alpha (i + 1)$ and $\beta (i) = \beta (i + 1)$. It follows immediately that $(u, v)$ is in the image of $s^{n - 1}_ i$. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)