Theorem 9.34 (0NQ3) · ScalingStacks

0NQ3

Theorem 9.34. Let ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ be a sequence of DHKS-saturated Waldhausen categories with factorization such that

\Ho(\mathrm{N}({\mathcal{A}})[W^{-1}][\Sigma^{-1}])\longrightarrow\Ho(\mathrm{N}({\mathcal{B}})[W^{-1}][\Sigma^{-1}])\longrightarrow\Ho(\mathrm{N}({\mathcal{C}})[W^{-1}][\Sigma^{-1}])

is a localization sequence of triangulated categories. Then the induced map

I\mspace{-6.mu}K({\mathcal{A}})\longrightarrow I\mspace{-6.mu}K({\mathcal{B}})\longrightarrow I\mspace{-6.mu}K({\mathcal{C}})

is a cofiber sequence of spectra.

0NQ4

Proof. This follows from the natural equivalence $I\mspace{-6.mu}K(-)\simeq I\mspace{-6.mu}K(\mathrm{N}(-)[W^{-1}][\Sigma^{-1}])$ and the fact that cofiber sequence

I\mspace{-6.mu}K(\mathrm{N}({\mathcal{A}})[W^{-1}][\Sigma^{-1}])\longrightarrow I\mspace{-6.mu}K(\mathrm{N}({\mathcal{B}})[W^{-1}][\Sigma^{-1}])\longrightarrow I\mspace{-6.mu}K(\mathrm{N}({\mathcal{C}})[W^{-1}][\Sigma^{-1}])

is a cofiber sequence because $I\mspace{-6.mu}K(-)$ is a localizing invariant. ∎