Kerodon

Exercise 6.3.7.18. Let $S$ be a simplicial set and let $\widetilde{S}$ be the smallest simplicial subset of $S \times \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$ which contains all simplices of the form $(\sigma , \tau )$, where $\tau $ is a nondegenerate simplicial subset of $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$ (that is, it corresponds to a strictly increasing sequence of nonnegative integers). Show that $\widetilde{S}$ is nonsingular, and that projection onto the first factor determines a universally localizing morphism $\widetilde{S} \twoheadrightarrow S$.