Notation 5.3.2.11 (The Mapping Simplex). Suppose we are given a diagram of simplicial sets
\[ X(0) \xrightarrow {f(1)} X(1) \xrightarrow { f(1) } X(2) \xrightarrow {f(3)} \cdots \xrightarrow {f(n)} X(n), \]
which we will identify with a functor $\mathscr {F}: [n] \rightarrow \operatorname{Set_{\Delta }}$. We denote the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ by $ \underset { \longrightarrow }{\mathrm{holim}}( X(0) \rightarrow \cdots \rightarrow X(n) )$, and refer to it as the mapping simplex of the diagram $\mathscr {F}$.