ScalingStacks

Proof (of 2.13). For $d\leq 0$ this follows directly from the definitions, and so we assume that $d\geq 1$. Let $K$ be a simplicial set. On the one hand,

	$\displaystyle\hom\left(K,\hom_{h_{d}\mathcal{C}}^{R}\left(X,Y\right)\right)$	$\displaystyle=$	$\displaystyle\hom_{\left(X,Y\right)}\left(J\left(K\right),h_{d}\mathcal{C}\right)$
		$\displaystyle=$	$\displaystyle\left[J\left(K\right)^{d-1},J\left(K\right)^{d},J\left(K\right)^{d+1};\mathcal{C}\right]_{\left(X,Y\right)},$

where subscript $\left(X,Y\right)$ indicates that we take only the subset of maps that restrict to $\left(X,Y\right)$ on $\partial\Delta^{1}\hookrightarrow J\left(K\right)$ (observe that this is independent of the representative as $\partial\Delta^{1}\subseteq J\left(K\right)^{d-1}$). On the other hand,

$\displaystyle\hom\left(K,h_{d-1}\hom_{\mathcal{C}}^{R}\left(X,Y\right)\right)$

$\displaystyle=$

$\displaystyle\left[K^{d-2},K^{d-1},K^{d};\hom_{\mathcal{C}}^{R}\left(X,Y\right)\right].$

We will argue that this last set is in natural bijection with the set

$$\left[J\left(K^{d-2}\right),J\left(K^{d-1}\right),J\left(K^{d}\right),\mathcal{C}\right]_{\left(X,Y\right)}.$$

First, by definition of the right mapping space we have a natural bijection between maps of the form $f\colon K^{d-1}\to\hom_{\mathcal{C}}^{R}\left(X,Y\right)$ and maps of the form $\overline{f}\colon J\left(K^{d-1}\right)\to\mathcal{C}$ restricting to $\left(X,Y\right)$ on $\partial\Delta^{1}\subseteq J\left(K^{d-1}\right)$. Second, $f$ extends to $K^{d}$ if and only if $\overline{f}$ extends to $J\left(K^{d}\right)$. Likewise, it is clear that two maps $f,g\colon K^{d-1}\to\hom_{\mathcal{C}}^{R}\left(X,Y\right)$ agree on $K^{d-2}$ if and only if the corresponding maps $\overline{f},\overline{g}\colon J\left(K^{d-1}\right)\to\mathcal{C}$ agree on $J\left(K^{d-2}\right)$. Hence, the only thing we need to show is that $f$ and $g$ are homotopic rel. $K^{d-2}$ if and only if $\overline{f}$ and $\overline{g}$ are homotopic rel. $J\left(K^{d-2}\right)$ and this follows from $\left(1\right)\iff\left(3\right)$ in 2.17. It remains to observe that for every simplicial set $K$ and every $d\geq 1$ we have a canonical isomorphism $J\left(K^{d-1}\right)\overset{\sim}{\longrightarrow}J\left(K\right)^{d}.$ Hence, we get a natural bijection

$$\hom\left(K,\hom_{h_{d}\mathcal{C}}^{R}\left(X,Y\right)\right)\simeq\hom\left(K,h_{d-1}\hom_{\mathcal{C}}^{R}\left(X,Y\right)\right)$$

and therefore an isomorphism $\alpha\colon\hom_{h_{d}\mathcal{C}}^{R}\left(X,Y\right)\simeq h_{d-1}\hom_{\mathcal{C}}^{R}\left(X,Y\right)$.