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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}

    Comments (1)

    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 4 comments on Section 14.18: Simplicial Methods.

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