at.algebraic topology - When do the Reedy and injective model category structures agree?

I've been thinking about this question too! It seems that showing that the Reedy and injective model structures agree for presheaves on $R$ boils down to having a well-behaved notion of "degeneracy".

Suppose you have a simplicial set $X$. Given an $n$-simplex $xin X([n])$, you say it is degenerate if there exists a $k<n$, a $k$-simplex $yin X([k])$, and a surjective map $sigma : [n]to [k]$ such that $X(sigma)(y)=x$. The simplex $x$ is non-degenerate if it is not degenerate.

Two facts:

1. If $Xsubseteq Y$ is an inclusion, and $xin X([n])$ is non-degenerate as a simplex of $X$, then it is also non-degenerate as a simplex of $Y$.




2. For every $xin X([n])$, there exists a unique pair $(y,sigma)$ consisting of a surjection $sigma:[n]to [k]$ and a non-degenerate $yin X([k])$ such that $X(sigma)(y)=x$.

These facts turn out to be what you need to show that the Reedy and injective model structures coincide on $mathrm{sSet}^{Delta^{op}}$; in particular, you can show that all monomorphisms are Reedy cofibrations (which is the hard part). If you replace $Delta$ with a Reedy category $R$ and make the appropriate definition of "degenerate" and "non-degenerate", conditions 1 and 2 are sufficient for injective=Reedy.

This leaves the question: why are 1 and 2 true for simplicial sets? I've known these facts for years, but only recently realized that I didn't know the proof! And then when I constructed a proof on my own, it turned out to be quite ugly. The good news is that there is in fact a very nice proof, due to Eilenberg and Zilber, and which you can find in section II.3 of Gabriel-Zisman, "Calculus of Fractions and Homotopy Theory".

I think these ideas generalize to $Theta_n$.