ScalingStacks

Proof. We must show that

$$\ell_{n}\colon(NC)_{n}\approx\Map_{s{\operatorname{\mathcal{S}}}}(F(n),NC)\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(\dot{F}(n),NC)$$

is a fibration for each $n\geq 0$. We have the following cases:

$n=0$:

$(NC)_{0}=\nerve(\iso C)$ is a Kan complex by (3.2).

$n=1$:

$\ell_{1}\colon(NC)_{1}\rightarrow(NC)_{0}\times(NC)_{0}$ is a simplicial covering space with discrete fiber, and thus is a fibration.

$n=2$:

$\ell_{2}$ is isomorphic to an inclusion of path-components, and so is a fibration.

$n\geq 3$:

$\ell_{n}$ is an isomorphism, and thus a fibration.

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