Lemma 12.19.2. Let \mathcal{A} be an abelian category. The category of filtered objects \text{Fil}(\mathcal{A}) has the following properties:
It is an additive category.
It has a zero object.
It has kernels and cokernels, images and coimages.
In general it is not an abelian category.
Proof. It is clear that \text{Fil}(\mathcal{A}) is additive with direct sum given by (A, F) \oplus (B, F) = (A \oplus B, F) where F^ p(A \oplus B) = F^ pA \oplus F^ pB. The kernel of a morphism f : (A, F) \to (B, F) of filtered objects is the injection \mathop{\mathrm{Ker}}(f) \subset A where \mathop{\mathrm{Ker}}(f) is endowed with the induced filtration. The cokernel of a morphism f : A \to B of filtered objects is the surjection B \to \mathop{\mathrm{Coker}}(f) where \mathop{\mathrm{Coker}}(f) is endowed with the quotient filtration. Since all kernels and cokernels exist, so do all coimages and images. See Example 12.3.13 for the last statement. \square
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