Abelian categories are not stable ∞-categories

Warning 4.2.4. Despite the similarity in their definitions, abelian categories are never stable \(\infty \)-categories: the loop functor \(\Omega \colon \Aa \to \Aa \) in a pointed 1-category \(\Aa \) is necessarily the zero-functor, hence is only an equivalence if \(\Aa \) is the trivial category.

Nevertheless, every abelian category \(\Aa \) embeds fully faithfully into a stable \(\infty \)-category \(\D (\Aa )\), called its derived \(\infty \)-category, to be introduced in Section 6.1 below. The inclusion \(\Aa \hookrightarrow \D (\Aa )\) sends short exact sequences in \(\Aa \) to exact sequences in \(\D (\Aa )\).