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Construction 5.2.6.3 (The Mapping Simplex). Suppose we are given a diagram of simplicial sets

\[ X(0) \xrightarrow {F(1)} X(1) \xrightarrow { F(2) } X(2) \xrightarrow {F(3)} \cdots \xrightarrow {F(n)} X(n), \]

which we identify with an $n$-simplex $\overrightarrow {X}$ of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{Set_{\Delta }})$. We let $M( \overrightarrow {X} )$ denote the fiber product $\Delta ^{n} \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }}) } \operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }})$, where $\operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }})$ is the universal mapping simplex of Notation 5.2.6.1. We will refer to $M( \overrightarrow {X} )$ as the mapping simplex of the diagram $\overrightarrow {X}$.