The Dold-Kan normalization functor reflects injectivity, surjectivity, and isomorphy.

Lemma 14.18.7. Let $\mathcal{A}$ be an abelian category. Let $f : U \to V$ be a morphism of simplicial objects of $\mathcal{A}$. If the induced morphisms $N(f)_ i : N(U)_ i \to N(V)_ i$ are injective for all $i$, then $f_ i$ is injective for all $i$. Same holds with “injective” replaced with “surjective”, or “isomorphism”.

Proof. This is clear from Lemma 14.18.6 and the definition of a splitting. $\square$

Comment #859 by Bhargav Bhatt on

Suggested slogan: The Dold-Kan normalization functor reflects injectivity, surjectivity, and isomorphy.

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