When do the Reedy and injective model category structures agree?

Question

Let $R$ be a Reedy category and consider the category $\mathcal{P}(R) = \mathbf{sSet}^{R^{\mathrm{op}}}$ of simplicial presheaves on $R$. When are the Reedy and injective model structures on $\mathcal{P}(R)$ the same? Are there useful sufficient conditions on $R$?

I know this is the case when all the morphisms of $R$ raise degree (I hope that's the right direction), and also when $R = \Delta$. I am specifically interested about the case where $R$ is $\Theta_n$, or a product of several copies of $\Theta_n$.

Answer 1

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 $x\in X([n])$, you say it is degenerate if there exists a $k<n$, a $k$-simplex $y\in X([k])$, 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 $X\subseteq Y$ is an inclusion, and $x\in X([n])$ is non-degenerate as a simplex of $X$, then it is also non-degenerate as a simplex of $Y$.

2. For every $x\in X([n])$, there exists a unique pair $(y,\sigma)$ consisting of a surjection $\sigma:[n]\to [k]$ and a non-degenerate $y\in X([k])$ 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 $\Theta_n$.

Answer 2

I complete the answer of Charles Rezk by some precise references: you may find the answer to your questions in my book Les préfaisceaux commes modèles des types d'homotopie, Astérisque 308 (2006). In chapter 8 of loc cit, I gave an axiomatic approach to this. Indeed, if $R$ is a skeletal category in the sense of Definition 8.1.1 in loc cit, and in which the automorphisms are trivial, then the class of monomorphisms in $\mathcal{P}(R)$ is generated by the set of boundary inclusions $\partial x\to x$, where $x$ runs over the family of representable precheaves; see Proposition 8.1.37. Now, these nice categories are closed under finite cartesian product (see Remark 8.1.7), from which you deduce easily that the Reedy model structure on the category of simplicial presheaves on $R$ coincides with the injective one. This applies of course to $R=\Theta_n$.

However, as I wrote this in French, I suspect some people might be happy to have a reference in English: Section 4 in this paper of Samuel B. Isaacson.