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 $B_ i$.