For a simplicial set $X$, is the category of non-degenerate simplices of $X$ a full subcategory of the category of simplices of $X$?

Question

I'm slightly confused, but I think you can help me. Let $X$ be a simplicial set. The category of simplices of $X$ and its subcategory of non-degenerate simplices are defined at http://ncatlab.org/nlab/show/category+of+simplices. Is this subcategory a full subcategory? I think it is, because I think that I can prove that it is, but as I don't really trust myself, I'd like to have your confirmation. What irritates me is that on the referred nLab page, and also in Hovey's "Model categories" (just before Lemma 3.1.4, which, by the way, had to be repaired slightly: see http://hopf.math.purdue.edu/Hovey/model-err.pdf), it is specifically emphasized that the morphisms be monomorphisms (or injective, respectively).

Answer 1

As far as I can tell you are correct. In fact, any map in the category of simplices coming out of a nondegenerate simplex must be injective. If $a:\Delta^n\to X$ is a nondegenerate simplex and $f:\Delta^n\to\Delta^m$ is a map to some other simplex $b:\Delta^m\to X$, factor $f=hg$ as a surjection followed by an injection. If $g$ is not the identity, then it makes $a$ a degeneracy of the simplex $bh$ in $X$. Thus $g=1$ and $f=h$ is injective.