ScalingStacks

Remark 13.4. If $V\in\Theta_{n}$ is of the form $V=\sigma(W)=([1],W)$ for some $W\in\Theta_{n-1}$, then any nondegenerate morphism $f\colon V\to C_{i}=([1],C_{i-1})$ is of the form $f=\sigma(g)$ for a unique nondegenerate $g\colon W\to C_{i-1}$.

More generally, the $\Theta$-construction gives rise, for each $[m]\in\Delta$, to functors

	$\displaystyle\sigma^{[m]}\colon\Theta_{n-1}^{\times m}$	$\displaystyle\to\Theta_{n}$
	$\displaystyle(o_{1},\dots,o_{m})$	$\displaystyle\mapsto({[m]};o_{1},\dots,o_{m}).$

In the case of $\sigma^{[1]}=\sigma$, we obtain functors

$$\sigma^{[m]}_{!}\colon\pre(\Theta_{n-1})^{\times m}\to\pre(\Theta_{n})$$

by left Kan extension in each variable. These functors were also considered by Rezk [34, § 4.4], and we adopt a similar notation: $\sigma^{[m]}_{!}(X_{1},\dots,X_{m})=({[m]};X_{1},\dots,X_{m})$.