Kerodon

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Construction 1.1.2.6 (The Boundary of $\Delta ^ n$). Let $n \geq 0$ be an integer. We define a simplicial set $(\operatorname{\partial \Delta }^ n): \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ by the formula

$( \operatorname{\partial \Delta }^{n} )( [m] ) = \{ \alpha \in \operatorname{Hom}_{\operatorname{{\bf \Delta }}}( [m], [n] ): \text{\alpha  is not surjective} \} .$

Note that we can regard $\operatorname{\partial \Delta }^ n$ as a simplicial subset of the standard $n$-simplex $\Delta ^ n$ of Construction 1.1.2.1. We will refer to $\operatorname{\partial \Delta }^ n$ as the boundary of $\Delta ^ n$.