# The Stacks Project

## Tag 017W

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}

Comment #859 by Bhargav Bhatt on July 26, 2014 a 4:24 pm UTC

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

There are also 3 comments on Section 14.18: Simplicial Methods.

## Add a comment on tag 017W

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).