Proposition 4.5 (0NKC) · ScalingStacks

0NKC

Proposition 4.5. Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be stable spectral categories. Then a spectral functor $f\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is a DK-equivalence if and only if

\Ho(\Omega^{\infty}f)\colon\Ho(\Omega^{\infty}{\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ho(\Omega^{\infty}{\mathcal{B}})

is an equivalence.

0NKD

Proof. Certainly essential surjectivity is determined on the level of the homotopy category, so it suffices to show that, for all pairs of objects $a$ and $b$ of ${\mathcal{A}}$ , $\pi_{n}\mathrm{Map}(a,b)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\pi_{n}\mathrm{Map}(fa,fb)$ for all integers $n$ whenever this is the case for $n=0$ . Since ${\mathcal{A}}$ and ${\mathcal{B}}$ are stable,

\pi_{0}\mathrm{Map}(\Sigma^{n}a,b)\cong\pi_{n}\mathrm{Map}(a,b)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\pi_{n}\mathrm{Map}(fa,fb)\cong\pi_{0}\mathrm{Map}(\Sigma^{n}fa,fb),

so this is immediate. ∎