## Tag `017W`

Chapter 14: Simplicial Methods > Section 14.18: Splitting simplicial objects

**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. $\square$

The code snippet corresponding to this tag is a part of the file `simplicial.tex` and is located in lines 2046–2059 (see updates for more information).

```
\begin{lemma}
\label{lemma-injective-map-simplicial-abelian}
\begin{slogan}
The Dold-Kan normalization functor reflects
injectivity, surjectivity, and isomorphy.
\end{slogan}
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''.
\end{lemma}
\begin{proof}
This is clear from Lemma \ref{lemma-splitting-abelian-category}
and the definition of a splitting.
\end{proof}
```

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