Stable ∞-category

Definition 4.2.1 (Stable \(\infty \)-category). An \(\infty \)-category \(C\) is called stable if:

(1)

The \(\infty \)-category \(C\) is pointed;

(2)

The \(\infty \)-category \(C\) admits fibers and cofibers;

(3)

A nullsequence \(X \xrightarrow {f} Y \xrightarrow {g} Z\) in \(C\) is a fiber sequence if and only if it is a cofiber sequence.

We refer to the nullsequences in (3) as exact sequences.¹ A functor \(F\colon C \to D\) between two stable \(\infty \)-categories is called exact if it preserves the zero object and sends exact sequences to exact sequences. Given two stable \(\infty \)-categories \(C\) and \(D\), we denote by \[ \Fun ^{\ex }(C,D) \quad \subseteq \quad \Fun (C,D) \] the full subcategory spanned by the exact functors.