Proposition 3.3 (0KEX) · ScalingStacks

[0KEX]

Proposition 3.3. Let $d\geq-1$ and let $p\colon\mathcal{C}\to\mathcal{D}$ be a functor, where $\mathcal{C}$ is an $\infty$ -category and $\mathcal{D}$ a $d$ -category.

(1)

If the functor $p\colon\mathcal{C}\to\mathcal{D}$ is an inner fibration, then so is $h_{d}\left(p\right)\colon h_{d}\left(\mathcal{C}\right)\to h_{d}\left(\mathcal{D}\right)=\mathcal{D}$ .
(2)

If in addition $f$ is a $p$ -coCartesian morphism in $\mathcal{C}$ , then $h_{d}\left(f\right)$ is $h_{d}\left(p\right)$ -coCartesian in $h_{d}\mathcal{C}$ .

[0KEY]

Proof. For $d=-1,0$ , both assertions are trivial to check and so we assume that $d\geq 1$ . The argument that $h_{d}\left(p\right)$ is an inner fibration is similar to the argument that $h_{d}\left(f\right)$ is coCartesian and so we shall prove them together. Using T.2.4.1.4, we need to consider the lifting problem

for some $m\geq 2$ and either

(1)

$0<i<m$ or
(2)

$i=0$ and $\Delta^{\left\{0,1\right\}}\subseteq\Lambda_{0}^{m}$ is mapped in $h_{d}\mathcal{C}$ to $h_{d}\left(f\right)$ .

For $m\geq d+3$ , we have $\mbox{sk}^{j}\Lambda_{i}^{m}=\mbox{sk}^{j}\Delta^{m}$ for all $j\leq d+1$ , and so the map

\hom\left(\Delta^{m},h_{d}\mathcal{C}\right)\to\hom\left(\Lambda_{i}^{m},h_{d}\mathcal{C}\right)

is a bijection and there is nothing to prove. For $m\leq d+2$ , we have $\Lambda_{i}^{m}=\mbox{sk}^{d+1}\Lambda_{i}^{m}$ , and so the map

\hom\left(\Lambda_{i}^{m},\mathcal{C}\right)\twoheadrightarrow\hom\left(\Lambda_{i}^{m},h_{d}\mathcal{C}\right)

is surjective, hence the map $\Lambda_{i}^{m}\to h_{d}\mathcal{C}$ factors through $\Lambda_{i}^{m}\to\mathcal{C}$ . Now, the functor $\mathcal{C}\to h_{d}\mathcal{C}$ identifies only homotopic morphisms (for $d\geq 1$ ); hence in (2) the image of $\Delta^{\left\{0,1\right\}}$ in $\mathcal{C}$ is coCartesian. Thus, in both cases we can solve the corresponding lifting problem in $\mathcal{C}$ , which induces a lift in the original square. ∎