Example 4.6.8.28 (01M0)

Example 4.6.8.28. Let $m$ and $n$ be nonnegative integers. By virtue of Remark 4.6.8.14, we can identify $m$-simplices of the simplicial set $\Phi ( \Delta ^ n )$ with the set $E( \Sigma (\Delta ^ n), m)$ defined in Notation 2.4.4.9. By definition, the elements of $E( \Sigma (\Delta ^ n), m)$ are given by pairs $(\sigma , \overrightarrow {I} )$, where $\sigma : \Delta ^ k \rightarrow \Sigma (\Delta ^ n)$ is a nondegenerate simplex of dimension $k > 0$ and $\overrightarrow {I} = ( I_0 \supseteq I_1 \supseteq \cdots \supseteq I_ m )$ is a chain of subsets of $[k]$ satisfying $I_0 = [k]$ and $I_ m = \{ 0, k\} $.

For each $k > 0$, there is a canonical bijection

\[ \{ \textnormal{Subsets $S \subseteq [n]$ of cardinality $k$} \} \simeq \{ \textnormal{Nondegenerate $k$-Simplices of $\Sigma (\Delta ^ n)$} \} , \]

which carries a subset $S$ to the $k$-simplex $\sigma _{S}$ given by the composite map

\[ \Delta ^ k \simeq \{ x\} \star \operatorname{N}_{\bullet }(S) \hookrightarrow \{ x\} \star \Delta ^ n \twoheadrightarrow \Sigma (\Delta ^ n). \]

For every such subset $S$, let $\iota _{S}: \operatorname{N}_{\bullet }(S) \hookrightarrow \Delta ^ k$ be the inclusion map. Then the construction

\[ ( \sigma _ S, \overrightarrow {I} ) \mapsto ( \sigma _{S}^{-1}(I_0) \supseteq \sigma _{S}^{-1}(I_1) \supseteq \cdots \supseteq \sigma _{S}^{-1}(I_ m) ) \]

induces a bijection from $E( \Sigma ( \Delta ^ n), m)$ to the collection of chains $\overrightarrow {S} = ( S_0 \supseteq S_1 \supseteq \cdots \supseteq S_ m )$ of subsets of $[n]$ which satisfy the following pair of conditions:

$(a)$: The set $S_ m$ contains exactly one element.
$(b)$: For $0 \leq i \leq m$, the unique element of $S_ m$ is the largest element of $S_ i$.

Let us henceforth use this bijection to identify $m$-simplices of $\Phi ( \Delta ^ n )$ with chains $\overrightarrow {S}$ satisfying $(a)$ and $(b)$. In these terms, the face and degeneracy operators for the simplicial set $\Phi ( \Delta ^ n ) = \Phi (\Delta ^ n)_{\bullet }$ can be described explicitly as follows:

For $0 \leq i \leq m$, the degeneracy operator $s^{m}_ i: \Phi ( \Delta ^ n)_{m} \rightarrow \Phi ( \Delta ^ n)_{m+1}$ is given by

\[ s^{m}_ i( S_0 \supseteq \cdots \supseteq S_ m ) = (S_0 \supseteq \cdots \supseteq S_{i-1} \supseteq S_{i} \supseteq S_ i \supseteq S_{i+1} \supseteq \cdots \supseteq S_ m) \]
For $0 \leq i < m$, the face operator $d^{m}_ i: \Phi ( \Delta ^ n)_{m} \rightarrow \Phi ( \Delta ^ n)_{m-1}$ is given by the construction

\[ d^{m}_ i( S_0 \supseteq \cdots \supseteq S_ m ) = (S_0 \supseteq \cdots \supseteq S_{i-1} \supseteq S_{i+1} \supseteq \cdots \supseteq S_ m). \]
For $m > 0$, the face operator $d^{m}_ m: \Phi ( \Delta ^ n)_{m} \rightarrow \Phi ( \Delta ^ n)_{m-1}$ is given by

\[ d^{m}_ m( S_0 \supseteq \cdots \supseteq S_ m ) = (S'_0 \supseteq S'_1 \supseteq \cdots \supseteq S'_{m-1} ), \]

where $S'_{i} = \{ j \in S_ i: j \leq \min (S_{m-1}) \} $.

See Remark 2.4.4.17.