Are simplicial sets equal if their non-degenerate simplices are the same?

Question

Let $\Delta$ be the simplex category and $\Delta^{inj}\subset \Delta$ be the non-full subcategory of injective morphisms. Then, a simplicial set $X:\Delta^{op}\to\text{Set}$ can be precomposed with the inclusion of $\Delta^{inj}$ to get a functor $X^{inj}:(\Delta^{inj})^{op}\to\textbf{Set}$ whose image are the non-degenerate morphisms of $X$.

Claim: Let $X,X'$ be simplicial sets such that $X^{inj} = X'^{inj}$. Then $X = X'$.

Proof: by Proposition 1.1.3.4 in Kerodon, any $n$-simplex $x:\Delta^n\to X$ can be factored as $\Delta^n\xrightarrow{f}\Delta^m\xrightarrow{y}X$, where $y$ is non-degenerate. Then $y\in X'$, since $X^{inj} = X'^{inj}$, hence $yf= x\in X'$. Thus $X\subset X'$. Analogously, $X'\subset X$ hence $X = X'$.

Is this correct? It would explain why the very frequent depictions of simplicial sets by their non-degenerate morphisms are mostly harmless.

Answer 1

It's not quite true that $X^\text{inj}$ (as you defined it) is the semismplicial set of non-degenerate simplices of $X$. In fact, it has exactly the same elements as $X$. On the other hand the collection of non-degenerate simplices does not always form a semisimplicial set: this is because there may be some non-degenerate simplex of $X$ that has a face that is degenerate.

There are some statements which are true. For example:

Let $X$ and $Y$ be simplicial sets, and let $f, g : X \to Y$ be morphisms of simplicial sets. If $f (x) = g (x)$ for all non-degenerate simplices $x$ of $X$, then $f = g$.

Another example:

Let $X$ and $Y$ be simplicial sets and let $f : X \to Y$ be a morphism of simplicial sets. If $f$ restricts to a bijection between non-degenerate simplices of $X$ and $Y$, then $f$ is an isomorphism of simplicial sets.