Notation 2.5.9.12 (00SP)

Notation 2.5.9.12. Let $I$ be a finite linearly ordered set of cardinality $n > 0$ and let $\operatorname{\raise {0.1ex}{\square }}^{I}$ denote the corresponding simplicial cube. For each element $i \in I$, the linear ordering on $I$ restricts to linear ordering on the subset $I \setminus \{ i\} $, which determines a fundamental chain

\[ [ \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ] \in \mathrm{N}_{n-1}( \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ; \operatorname{\mathbf{Z}}). \]

We will write $[ \{ 0\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ] \in \mathrm{N}_{n-1}( \operatorname{\raise {0.1ex}{\square }}^{I}; \operatorname{\mathbf{Z}})$ for the image of the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ]$ under the inclusion of simplicial sets

\[ \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \simeq \{ 0\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \hookrightarrow \Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \simeq \operatorname{\raise {0.1ex}{\square }}^{I}. \]

Similarly, we write $[ \{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ] \in \mathrm{N}_{n-1}( \operatorname{\raise {0.1ex}{\square }}^{I}; \operatorname{\mathbf{Z}})$ for the image of the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ]$ under the inclusion

\[ \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \simeq \{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \hookrightarrow \Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \simeq \operatorname{\raise {0.1ex}{\square }}^{I}. \]