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?
Best Answer
I realize I'm quite late for this post, but I believe it deserves an answer as the theorem is so often implicitly used when constructing maps between simplicial sets. Rather than addressing OP's original question, I try to give a complete proof here.
Denote by $X^{\text{nd}}_\bullet$ the semi-simplicial set of non-degenerate simplices of $X_\bullet$ and $\varphi: X^{\text{nd}}_\bullet \to Y_\bullet$ a map of semi-simplicial complex, that is, a graded map compatible with face maps. We extend $\varphi$ to a map of simplicial complex $\varphi: X_\bullet \to Y_\bullet$ as follows.
For each degenerate $n$-simplex $\Delta^n_\bullet \to X_\bullet$, there is a unique factorization $f\circ \alpha_*: \Delta^n_\bullet \to \Delta^k_\bullet \to X_\bullet$, where the first map is induced by a surjective map $\alpha: [n] \to [k]$ and the second map $f$ defines a non-degenerate $k$-simplex (this is proved in the subsection on skeleton filtration in Kerodon). We set $\varphi(f \circ \alpha_*) = \varphi(f) \circ \alpha_*$.
This is the only way to extend $\varphi$ so we have uniqueness. Note also that the extension respects degeneracy maps, as $f\circ (\alpha_* \circ s^i)$ is the unique factorization of $(f\circ \alpha_*) \circ s^i$. To check compatibility with face maps, we write $\alpha_*$ as compositions of (co)degeneracy maps:
$$\alpha_* = s^{i_m} \circ \cdots \circ s^{i_1}.$$
By simplicial identities, $$\alpha_* \circ d^j = s^{i_m} \circ \cdots \circ s^{i_1} \circ d^j = d^{j'} \circ s^{i'_m} \circ \cdots \circ s^{i'_1} = d^{j'} \circ \alpha'_*,$$
where $\alpha': [n-1] \to [k-1]$ is surjective. (Reminder: This is not true. As PaulTaylors pointed out in the comment, we may encounter cases that $s^k \circ d^l = \text{id}$, resulting $\alpha_* \circ d^j = \alpha'_*$, where $\alpha': [n-1] \to [k]$ is surjective.) We thus have \begin{align*} \varphi(d_j(f \circ \alpha_*)) =& \varphi(f \circ \alpha_*\circ d^j) \\ =& \varphi((f \circ d^{j'})\circ \alpha'_*) \\ =&\varphi(f \circ d^{j'}) \circ \alpha'_*\\ =&\varphi(f) \circ d^{j'} \circ \alpha'_*\\ =&\varphi(f) \circ \alpha_*\circ d^j\\ =&d_j(\varphi(f) \circ \alpha_*), \end{align*}
where in the third last equality we use that $\varphi$ on non-degenerate simplices is compatible with faces maps.