# Simplicial morphisms from degenerate simplices

Frequently I read the following statement on morphisms of simplicial sets:

Suppose $X$ and $Y$ are simplicial sets, $f: X \rightarrow Y$ is a simplicial morphism and we write $X_n$ as well as $Y_n$ and $f_n$ for the dimension $n$ parts. Moreover suppose that there is a $n_0 \in \mathbb{N}_0$ such that $X$ has only degenerate simplices $x \in X_n$ for all $n \geq n_0$. (That is each such element is in the image of at least one degeneracy map)

Then any simplicial morphism is determined by its dimension $\leq n$ -parts, that is each $f$ is determined by all the $f_m$'s for all $m \leq n$.

Unfortunately I never read a proof for this and hence I would like to know how to proof it. I guess that the Eilenberg Zilber lemma that gives a decomposition of any simplex $x$ into a non degenerate simplex a right action of a monotonic map in the simplex category is the basic idea in the proof, but I don't realy know how to work it out.

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Since this is my first post here, I don't know if it is custom here to ask for complete proofs, but at least a reference to a proof would be great.

Thanks, Marc

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The properties that a simplicial morphism must satisfy include $f_ns_i=s_if_{n-1}$, so this property follows immediately. – Fernando Muro Mar 7 '12 at 21:38
"this property" means that in your question – Fernando Muro Mar 7 '12 at 22:37
If you write this as an answer I can flag it as solved. – Mark.Neuhaus Mar 9 '12 at 1:50