The Stacks project

Lemma 14.22.1. Let $\mathcal{A}$ be an abelian category.

  1. The categories $\text{Simp}(\mathcal{A})$ and $\text{CoSimp}(\mathcal{A})$ are abelian.

  2. A morphism of (co)simplicial objects $f : A \to B$ is injective if and only if each $f_ n : A_ n \to B_ n$ is injective.

  3. A morphism of (co)simplicial objects $f : A \to B$ is surjective if and only if each $f_ n : A_ n \to B_ n$ is surjective.

  4. A sequence of (co)simplicial objects

    \[ A \xrightarrow {f} B \xrightarrow {g} C \]

    is exact at $B$ if and only if each sequence

    \[ A_ i \xrightarrow {f_ i} B_ i \xrightarrow {g_ i} C_ i \]

    is exact at $B_ i$.

Proof. Pre-additivity is easy. A final object is given by $U_ n = 0$ in all degrees. Existence of direct products we saw in Lemmas 14.6.2 and 14.9.2. Kernels and cokernels are obtained by taking termwise kernels and cokernels. $\square$

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