ScalingStacks

Proof. Denote by $j$ any Yoneda embedding (the context will always be made clear). Let $K$ denote the simplicial set as in 12.2, which we regard as a simplicial space that is discrete in each degree. This is a pushout along an inclusion, hence this is also a (homotopy) pushout in the $\infty$-category of simplicial spaces. Now let $T$ be the strongly saturated class of morphisms of $\Fun(\mathrm{N}\Delta^{\mathrm{op}},\CSS(\Delta^{\!\times n-1}))$ generated by the three sets

	$\displaystyle\{j([0],\mathbf{m})\to j(\mathbf{0})\ |\ \mathbf{m}\in\Delta^{\!\times n-1}\},$
	$\displaystyle\{\mathrm{Segal}_{\Delta}\boxtimes\mathbf{m}\ |\ \mathbf{m}\in\Delta^{\!\times n-1}\},$
	$\displaystyle\{c_{n-1}(K)\to j(\mathbf{0})\},$

One deduces immediately that a simplicial object of $\CSS(\Delta^{\!\times n-1})$ is a Segal space if and only if it is local with respect to each of the first two sets of morphisms. To show that $\CSS(\Delta^{\!\times n})$ coincides with the localization $T^{-1}\Fun(\mathrm{N}\Delta^{\mathrm{op}},\CSS(\Delta^{\!\times n-1}))$, it is enough to show that a $1$-fold Segal space $X$ is complete if and only if the natural map

$$X_{0}\to\map(K,X)$$

is an equivalence. By the Yoneda lemma, our claim is just a restatement of [34, Proposition 10.1]. ∎