Construction 3.6.2.6 (013J)

Construction 3.6.2.6. Let $S$ be a finite subset of the unit interval $[0,1]$, and assume that $0,1 \in S$. For each $n \geq 0$, we let $| \Delta ^ n |_{S}$ denote the subset of the topological $n$-simplex

\[ | \Delta ^ n | = \{ (t_0, \ldots , t_ n) \in \operatorname{\mathbf{R}}_{\geq 0}^{n+1}: t_0 + t_1 + \cdots + t_ n = 1 \} \]

consisting of those tuples $(t_0, t_1, \ldots , t_ n)$ having the property that each of the partial sums $t_0 + t_1 + \cdots t_ i$ belongs to $S$. Note that these subsets are stable under the coface and codegeneracy operators of the cosimplicial topological space $| \Delta ^{\bullet } |$, so we can regard the construction $[n] \mapsto | \Delta ^ n |_{S}$ as a cosimplicial set.

By virtue of Proposition 1.2.3.15, the functor

\[ \operatorname{Set}\rightarrow \operatorname{Set_{\Delta }}\quad \quad (Y \mapsto ([n] \mapsto \operatorname{Hom}_{\operatorname{Set}}( | \Delta ^{n} |_{S}, Y))) \]

admits a left adjoint, which we will denote by $| \bullet |_{S}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}$ and refer to as the $S$-partial geometric realization. Concretely, this functor carries a simplicial set $X$ to the colimit $|X|_{S} = \varinjlim _{ \Delta ^{n} \rightarrow X} | \Delta ^{n} |_{S}$, where the colimit is indexed by the category of simplices $\operatorname{{\bf \Delta }}_{X}$ of Construction 1.1.3.9.