The Stacks project

Definition 12.10.5. Let $\mathcal{A}$, $\mathcal{B}$ be abelian categories. Let $F : \mathcal{A} \to \mathcal{B}$ be an exact functor. Then the full subcategory of objects $C$ of $\mathcal{A}$ such that $F(C) = 0$ is called the kernel of the functor $F$, and is sometimes denoted $\mathop{\mathrm{Ker}}(F)$.

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View Definition 12.10.5 as pdf