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Tag 017S

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

Lemma 14.18.3. Let $f : U \to V$ be a morphism of simplicial sets. Suppose that (a) the image of every nondegenerate simplex of $U$ is a nondegenerate simplex of $V$ and (b) no two nondegenerate simplices of $U$ are mapped to the same simplex of $V$. Then $f_n$ is injective for all $n$. Same holds with ''injective'' replaced by ''surjective'' or ''bijective''.

Proof. Under hypothesis (a) we see that the map $f$ preserves the disjoint union decompositions of the splitting of Lemma 14.18.2, in other words that we get commutative diagrams $$ \xymatrix{ \coprod\nolimits_{\varphi : [n] \to [m]\text{ surjective}} N(U_m) \ar[r] \ar[d] & U_n \ar[d] \\ \coprod\nolimits_{\varphi : [n] \to [m]\text{ surjective}} N(V_m) \ar[r] & V_n. } $$ And then (b) clearly shows that the left vertical arrow is injective (resp. surjective, resp. bijective). $\square$

    The code snippet corresponding to this tag is a part of the file simplicial.tex and is located in lines 1812–1822 (see updates for more information).

    \begin{lemma}
    \label{lemma-injective-map-simplicial-sets}
    Let $f : U \to V$ be a morphism of simplicial sets.
    Suppose that (a) the image of every nondegenerate simplex of
    $U$ is a nondegenerate simplex of $V$ and (b)
    no two nondegenerate simplices of $U$ are mapped
    to the same simplex of $V$.
    Then $f_n$ is injective for all $n$.
    Same holds with ``injective'' replaced by
    ``surjective'' or ``bijective''.
    \end{lemma}
    
    \begin{proof}
    Under hypothesis (a) we see that the map $f$ preserves
    the disjoint union decompositions of the splitting
    of Lemma \ref{lemma-splitting-simplicial-sets}, in other words
    that we get commutative diagrams
    $$
    \xymatrix{
    \coprod\nolimits_{\varphi : [n] \to [m]\text{ surjective}}
    N(U_m)
    \ar[r] \ar[d] &
    U_n \ar[d] \\
    \coprod\nolimits_{\varphi : [n] \to [m]\text{ surjective}}
    N(V_m)
    \ar[r] &
    V_n.
    }
    $$
    And then (b) clearly shows that the left vertical arrow is
    injective (resp.\ surjective, resp.\ bijective).
    \end{proof}

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