ScalingStacks

Proof. The set $S_{00}$ consists of the union of four subsets of maps, corresponding to the four families of fundamental pushouts of types (a), (b), (c), and (d) in Axiom (C.3). The second and last subsets corresponding to the (b) and (d) families pullback to morphisms which are contained in the generating set of $T_{\Theta_{n}}$. Thus it remains to prove the that the same holds for the remaining families (a) and (c). In particular, we wish to show that for each $0\leq i\leq n$, each $0\leq j,k\leq n-i$, and every nondegenerate morphism $C_{i+j}\to C_{i}$ and $C_{i+k}\to C_{i}$, the natural morphism

(13.2.1)		$\displaystyle f(C_{i+j}\cup^{C_{i}}C_{i+k}))\cup^{f(\sigma^{i+1}(C_{j-1}\times C_{k-1}))}(f(C_{i+k}\cup^{C_{i}}$	$\displaystyle C_{i+j})$
		$\displaystyle\to f(C_{i+j}\times_{C_{i}}C_{i+k}),$

is contained in $T_{\Theta_{n}}$ where the pushout is formed as in Notation 6.5.

In fact a stronger statement holds (cf. [34, Proposition 4.9]). For each object $o\in\Theta_{n}$ we have a natural bijection of sets

$$\hom(o,C_{i+j}\times_{C_{i}}C_{i+k})\cong\hom(o,C_{i+j}\cup^{C_{i}}C_{i+k}))\cup^{\hom(o,C_{i+m})}\hom(o,C_{i+k}\cup^{C_{i}}C_{i+j}).$$

Thus Eq. 13.2.1 is in fact an equivalence in the presheaf category $\pre(\Theta_{n})$. In particular the family (c) pulls back to a family of equivalences, which are hence contained in $T_{\Theta_{n}}$. An virtually identical argument applies the family (a), which also consists of morphisms pulling back to equivalences of presheaves. ∎