Levelwise detection of isomorphisms in limit categories

Proof. Let \(C\) be a stable \(\infty \)-category. If \(I\) is a finite \(\infty \)-category, then \(C\) has \(I\)-indexed colimits by the previous corollary, hence there is a functor \(\colim \colon \Fun (I,C) \to C\). Its domain is also stable and \(\colim \) preserves finite colimits, because colimits commute with colimits. So the functor is exact and hence preserves finite limits as well.

Conversely, assume that \(C\) is finitely complete and cocomplete and that finite limits commute with finite colimits in \(C\). Considering the empty category, the fact that empty limits commute with empty colimits expresses that \(C\) is pointed. To show that \(C\) is stable, consider for every \(X \in C\) the following diagram: \begin {equation*}

\end {equation*} Taking pushouts of the rows we get the diagram \(0 \to \Sigma X \leftarrow 0\) whose pullback is \(\Omega \Sigma X\). If instead we take pullbacks of the columns, we get \(X \leftarrow X \to X\), whose pushout is \(X\). Pullbacks commuting with pushouts tell us then the canonical map \(X \to \Omega \Sigma X\) is an equivalence. A similar argument shows that \(\Sigma \Omega X \to X\) is an equivalence, so that \(C\) is stable by Theorem 4.2.2. □