TITLE: When do the Reedy and injective model category structures agree?
QUESTION [10 upvotes]: 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 $\Theta_n$.

REPLY [9 votes]: 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.