ScalingStacks

Proposition 2.9. A spectral functor $F\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is

• •

  a triangulated equivalence if and only if the induced derived functor

  $$\mathbb{L}F_{!}\colon{\mathcal{D}}_{\tri}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}_{\tri}({\mathcal{B}})$$

  is an equivalence of (triangulated) categories,

• •

  a Morita equivalence if and only if the induced derived functor

  $$\mathbb{L}F_{!}\colon{\mathcal{D}}_{\perf}({\mathcal{A}})\longrightarrow{\mathcal{D}}_{\perf}({\mathcal{B}})$$

  is an equivalence of (triangulated) categories.