ScalingStacks

Proof. By construction, ${\mathcal{B}}/{\mathcal{A}}\subseteq{\mathcal{B}}$ is the full subcategory on those objects $b$ such that $\map(a,b)\simeq*$ for all objects $a$ in the image of ${\mathcal{A}}$. This shows that, as full subcategories of $\Ho({\mathcal{B}})$, $\Ho({\mathcal{B}}/{\mathcal{A}})\subseteq\Ho({\mathcal{B}})/\Ho({\mathcal{A}})$. Conversely, if $b$ is in $\Ho({\mathcal{B}})/\Ho({\mathcal{A}})$, then $\pi_{0}\map(a,b)\simeq\ast$ for each object $a$ in the image of ${\mathcal{A}}$, and we claim that in fact $\map(a,b)\simeq*$. Indeed, ${\mathcal{A}}$ is a stable subcategory of ${\mathcal{B}}$, so that $\pi_{n}\map(a,b)\simeq\pi_{0}\map(\Sigma^{n}a,b)\simeq\ast$. Hence $\Ho({\mathcal{B}})/\Ho({\mathcal{A}})\subseteq\Ho({\mathcal{B}}/{\mathcal{A}})$ as well. ∎