Construction 3.3.6.1 (00ZB): The $\operatorname{Ex}^{\infty }$ Functor

Construction 3.3.6.1 (The $\operatorname{Ex}^{\infty }$ Functor). For every nonnegative integer $n$, we let $\operatorname{Ex}^{n}$ denote the $n$-fold iteration of the functor $\operatorname{Ex}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ of Construction 3.3.2.5, given inductively by the formula

\[ \operatorname{Ex}^{n}(X) = \begin{cases} X & \text{ if } n = 0 \\ \operatorname{Ex}( \operatorname{Ex}^{n-1}(X) ) & \text{ if } n > 0. \end{cases} \]

For every simplicial set $X$, we let $\operatorname{Ex}^{\infty }(X)$ denote the colimit of the diagram

\[ X \xrightarrow { \rho _{X} } \operatorname{Ex}(X) \xrightarrow { \rho _{ \operatorname{Ex}(X) } } \operatorname{Ex}^2(X) \xrightarrow { \rho _{ \operatorname{Ex}^2(X)} } \operatorname{Ex}^3(X) \rightarrow \cdots , \]

where each $\rho _{ \operatorname{Ex}^{n}(X) }$ denotes the comparison map of Construction 3.3.4.3, and we let $\rho ^{\infty }_{X}: X \rightarrow \operatorname{Ex}^{\infty }(X)$ denote the tautological map. The construction $X \mapsto \operatorname{Ex}^{\infty }(X)$ determines a functor $\operatorname{Ex}^{\infty }$ from the category of simplicial sets to itself, and the construction $X \mapsto \rho _{X}^{\infty }$ determines a natural transformation of functors $\operatorname{id}_{ \operatorname{Set_{\Delta }}} \rightarrow \operatorname{Ex}^{\infty }$.