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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