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Notation (Cubes as Simplicial Sets). Let $I$ be a set. We let $\operatorname{\raise {0.1ex}{\square }}^{I}$ denote the simplicial set given by the product $\prod _{i \in I} \Delta ^1$. We will refer to $\operatorname{\raise {0.1ex}{\square }}^{I}$ as the $I$-cube. Equivalently, we can describe $\operatorname{\raise {0.1ex}{\square }}^{I}$ as the nerve of the power set $P(I) = \{ I_0 \subseteq I \} $, where we regard $P(I)$ as partially ordered with respect to inclusion.

In the special case where $I$ is the set $\{ 1, 2, \ldots , n \} $ for some nonnegative integer $n$, we will denote the simplicial set $\operatorname{\raise {0.1ex}{\square }}^{I}$ by $\operatorname{\raise {0.1ex}{\square }}^{n}$ and refer to it as the standard $n$-cube.