Proposition 16.5.7. Let \(C\) be a pointed \(\infty \)-category with finite colimits and let \(D\) be an \(\infty \)-category with finite limits. Then \(\Exc _*(C,D)\) is stable.
Proof. By replacing \(D\) with \(D_*\), we may assume \(D\) is pointed. The \(\infty \)-category \(\Exc _*(C,D)\) has finite limits computed pointwise, and reduced excisive functors are closed under such limits. It is also pointed. By Theorem 4.2.2, it remains to show that its loop functor is an equivalence.
The loop functor is computed pointwise: \(\Omega (F)=\Omega _D\circ F\). Define a shift functor by \(\Shift (F):=F\circ \Sigma _C\). Since \(\Sigma _C\) preserves zero objects and pushout squares, \(\Shift (F)\) is again reduced excisive. Applying an excisive functor \(F\) to the pushout square defining suspension gives natural isomorphisms \[ F \simeq \Omega _DF\Sigma _C = \Omega (\Shift F) = \Shift (\Omega F), \] so \(\Omega \) and \(\Shift \) are inverse equivalences. □
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