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Notation 5.2.6.1 (The Universal Mapping Simplex). Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets and let $\operatorname{N}_{\bullet }( \operatorname{Set_{\Delta }})$ denote its nerve. For each $n \geq 0$, we can identify $\operatorname{N}_{n}(\operatorname{Set_{\Delta }})$ with the set of functors from the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $ to the category $\operatorname{Set_{\Delta }}$. In what follows, we will typically denote such a functor by $\overrightarrow {X}$ and will denote its value on an integer $0 \leq i \leq n$ by $X(i)$. Let $\operatorname{N}^{+}_{n}(\operatorname{Set_{\Delta }})$ denote the set of pairs $( \overrightarrow {X}, \sigma )$, where $\overrightarrow {X}: [n] \rightarrow \operatorname{Set_{\Delta }}$ is a functor and $\sigma $ is an $n$-simplex of the simplicial set $X(0)$. Every nondecreasing function $\alpha : [m] \rightarrow [n]$ determines a map $\alpha ^{\ast }: \operatorname{N}^{+}_{n}(\operatorname{Set_{\Delta }}) \rightarrow \operatorname{N}^{+}_{m}(\operatorname{Set_{\Delta }})$, which carries $( \overrightarrow {X}, \sigma )$ to the pair $( \overrightarrow {X} \circ \alpha , \sigma ' )$ where $\sigma '$ denotes the composite map

\[ \Delta ^{m} \xrightarrow {\alpha } \Delta ^{n} \xrightarrow {\sigma } X(0) \rightarrow X( \alpha (0) ). \]

The construction $[n] \mapsto \operatorname{N}^{+}_{n}( \operatorname{Set_{\Delta }})$ determines a simplicial set $\operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }})$, which we will refer to as the universal mapping simplex.