ScalingStacks

Proof. Let $d^{0}H(k-1)\subset C(k)$ denote the image of $H(k-1)$ in $C(k)$ induced by the map $d^{0}\colon F(k-1)\rightarrow F(k)$. There is a square

of subobjects of $C(k)$; we need to show that the inclusion map $d^{0}H(k-1)\cup\alpha^{0}F(1)\rightarrow C(k)$ of the union of these subobjects is a weak equivalence in the Segal space model category structure.

Now $C(k)$ can be written as a colimit of the poset of subcomplexes each of which

1. (1)

   are isomorphic to $F(\ell)$ for some $\ell<k$, and

2. (2)

   include $0,1\in F(k)_{0}$.

Straightforward calculation shows that the intersection of $d^{0}H(k-1)\cup\alpha^{0}F(1)$ with each of the objects $F(\ell)$ in the above diagram is a cover of $F(\ell)$. ∎