I'm studying "Introduction to $\infty$-categories", by Markus Land, and I found myself in need of constructing simplicial maps from their values on non singular simplices.
This is what I'd like to prove:
Theorem (?)
Suppose that X and Y are simplicial sets, and that we are given maps
$\varphi:\{e\in X_k| \text{e is non degenerate}\}\rightarrow Y_k$
such that the following condition holds:
a) Let e and e' be non degenerate simplices of X such that e=X(f)(e'), where $f$ an injective morphism in the simplex category $\Delta$. Then $\varphi(e)=Y(f)\varphi(e')$.
Then there is a unique extension of these maps to a simplicial map $\varphi:X\rightarrow Y$.
I already proved the uniqueness of $\varphi$ under condition a). The extension is necessarily defined by $\varphi(a)=Y(\sigma)(\varphi(e))$ if $a=X(\sigma)(e),$ with $e$ a non degenerate simplex of X and $\sigma$ a surjective morphism in $\Delta$.
I also proved that the existence of $\varphi$ is actually equivalent to condition a) plus the following condition:
b) Suppose $e,e'$ are non degenerate simplices of X, and suppose that $s_{j_h}\dots s_{j_1}(e')=d_{i_k}\dots d_{i_1}(e)$, where $s_{j_l}$ and $d_{i_r}$ are degeneracy maps or face maps of X, respectively. Then $s_{j_1}\dots s_{j_1}(\varphi(e'))=d_{i_k}\dots d_{i_1}(\varphi(e))$ as simplices of Y.
It turns out that we can also just assume that k=1 in b).
I think that it would be nice if b) followed from condition a) alone, but I'm not so sure this is true.
So, I'd like to ask: does condition b) actually follow from a) alone?