Notation 2.4.5.5 (Subsets of the $I$-Cube). Let $I$ be a finite set and let $\operatorname{\raise {0.1ex}{\square }}^{I}$ denote the $I$-cube of Notation 2.4.5.2. For each element $i \in I$, we can identify $\operatorname{\raise {0.1ex}{\square }}^{I}$ with the product $\Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} }$. Using this identification, we obtain simplicial subsets
which we will refer to as faces of $\operatorname{\raise {0.1ex}{\square }}^{I}$. The (disjoint) union of these two faces is another simplicial subset of $\operatorname{\raise {0.1ex}{\square }}^{I}$, which we can identify with the product $\partial \Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} }$.
We let $\operatorname{\partial \raise {0.1ex}{\square }}^{I}$ denote the simplicial subset of $\operatorname{\raise {0.1ex}{\square }}^{I}$ given by the union
of all its faces. We will refer to $\operatorname{\partial \raise {0.1ex}{\square }}^{I}$ as the boundary of the $I$-cube $\operatorname{\raise {0.1ex}{\square }}^{I}$.
For $i \in I$, we let $\boldsymbol {\sqcap }^{I}_{i} \subseteq \operatorname{\raise {0.1ex}{\square }}^{I}$ denote the simplicial subset of $\operatorname{\raise {0.1ex}{\square }}^{I}$ given by the union of the face $(\{ 0\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } )$ with $\bigcup _{j \in I \setminus \{ i\} } ( \partial \Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ j\} })$. Similarly, we let $\boldsymbol {\sqcup }^{I}_{i}$ denote the simplicial subset of $\operatorname{\raise {0.1ex}{\square }}^{I}$ given by the union of the face $(\{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } )$ with $\bigcup _{j \in I \setminus \{ i\} } ( \partial \Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ j\} })$. We will refer to the simplicial subsets $\boldsymbol {\sqcap }^{I}_{i}, \boldsymbol {\sqcup }^{I}_{i} \subseteq \operatorname{\raise {0.1ex}{\square }}^{I}$ as hollow $I$-cubes.
In the special case where $I = \{ 1, \ldots , n \} $ for some nonnegative integer $n$, we will denote the simplicial sets $\operatorname{\partial \raise {0.1ex}{\square }}^{I}$, $\boldsymbol {\sqcap }^{I}_{i}$, and $\boldsymbol {\sqcup }^{I}_{i}$ by $\operatorname{\partial \raise {0.1ex}{\square }}^{n}$, $\boldsymbol {\sqcap }^{n}_{i}$, and $\boldsymbol {\sqcup }^{n}_{i}$, respectively.