The Stacks project

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) the restriction of $f$ to a map from the set of nondegenerate simplices of $U$ to the set of nondegenerate simplices of $V$ is injective. 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). $\square$

Comments (2)

Comment #4265 by searial on

It seems that the condition (b) also should be replaced appropriately when “injective” replaced by “surjective” or “bijective”.

There are also:

  • 6 comment(s) on Section 14.18: Splitting simplicial objects

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