Corollary 16.5.6. The following conditions are equivalent for a reduced functor \(F\colon C \to D\) between stable \(\infty \)-categories:
- (1)
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\(F\) is an exact functor;
- (2)
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\(F\) commutes with loop objects;
- (3)
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\(F\) commutes with suspensions.
Proof. Since \(C\) and \(D\) are stable, the functor \(F\) is exact if and only if it is excisive. By Proposition 16.5.5, this holds if and only if the canonical map \(F(X)\to \Omega _DF(\Sigma _CX)\) is an isomorphism for every \(X\in C\). By adjunction, this is equivalent to the canonical comparison \(\Sigma _DF(X)\to F(\Sigma _CX)\) being an isomorphism. Thus exactness is equivalent to preservation of suspensions, and preservation of loops is equivalent because suspension and loops are inverse equivalences in both categories. □
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