Proposition 5.9 (0NLT) · ScalingStacks

0NLT

Proposition 5.9. Let ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ be a fully faithful inclusion of presentable stable $\infty$ -categories. Then the natural map $\Ho({\mathcal{B}})/\Ho({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho({\mathcal{B}}/{\mathcal{A}})$ is an equivalence.

0NLU

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. ∎