Proposition 5.14 (0NM0) · ScalingStacks

0NM0

Proposition 5.14. Let ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ be a fully faithful and $\kappa$ -small colimit preserving functor of $\kappa$ -cocomplete small 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. In other words, the functor $\Ho(-)$ preserves quotients of fully faithful functors.

0NM1

Proof. We have equivalences

\Ho(\Ind_{\kappa}({\mathcal{B}}))/\Ho(\Ind_{\kappa}({\mathcal{A}}))\simeq\Ho(\Ind_{\kappa}({\mathcal{B}})/\Ind_{\kappa}({\mathcal{A}}))\simeq\Ho(\Ind_{\kappa}({\mathcal{B}}/{\mathcal{A}})),

where the first equivalence follows from proposition 5.9 and the last equivalence follows from the fact that $\Ind_{\kappa}$ preserves cofibers. We therefore obtain a commutative (up to natural isomorphism) square

where the right vertical map is fully faithful and the bottom map is an equivalence.

To see that the top vertical map is fully faithful, we use Neeman’s generalization of Thomason’s localization theorem (see [61, 4.4.9] or [62]) to show that the left vertical map is fully faithful. First, since $\Ind_{\kappa}({\mathcal{A}})$ and $\Ind_{\kappa}({\mathcal{B}})$ are presentable, the criterion of [50] (characterizing well-generated triangulated categories) and [53, 1.4.5.2] imply that $\Ho(\Ind_{\kappa}({\mathcal{B}}))$ is well-generated and (since the map $\Ind_{\kappa}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind_{\kappa}({\mathcal{B}})$ is fully-faithful) the image of $\Ho(\Ind_{\kappa}({\mathcal{A}}))$ is a localizing subcategory generated by a small set of objects. Applying the form of Neeman’s theorem proved by Krause in [51, 7.2.1] now implies that

\Ho\Ind_{\kappa}({\mathcal{B}})^{\kappa}/\Ho\Ind_{\kappa}({\mathcal{A}})^{\kappa}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho\Ind_{\kappa}({\mathcal{B}})/\Ho\Ind_{\kappa}({\mathcal{A}})

is a fully faithful map. Since [53, 1.4.5.1] implies that there is an equivalence $\Ho(\Ind_{\kappa}({\mathcal{A}})^{\kappa})\simeq\Ho(\Ind_{\kappa}({\mathcal{A}}))^{\kappa}$ and $\Ind_{\kappa}({\mathcal{A}})^{\kappa}\simeq{\mathcal{A}}$ up to idempotent completion (and similarly for ${\mathcal{B}}$ ), we conclude that the left vertical map is fully faithful.

Finally, this map is essentially surjective because there is a commutative (up to natural isomorphism) triangle

such that both maps from $\Ho({\mathcal{B}})$ are essentially surjective. ∎