The Stacks project

Definition 13.6.5. Let $\mathcal{D}$ be a (pre-)triangulated category.

1. Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor. The kernel of $F$ is the strictly full saturated (pre-)triangulated subcategory described in Lemma 13.6.2.

2. Let $H : \mathcal{D} \to \mathcal{A}$ be a homological functor. The kernel of $H$ is the strictly full saturated (pre-)triangulated subcategory described in Lemma 13.6.3.

These are sometimes denoted $\mathop{\mathrm{Ker}}(F)$ or $\mathop{\mathrm{Ker}}(H)$.