# Can Reedy cofibrations be monomorphisms?

In Hirschhorn's "Model Categories and Their Localizations", section 15.7, there's the following corollary to the preceding Proposition:

$$\mathbb{Corollary}$$ 15.7.2 If $$\mathfrak{M}$$ is a cellular model category and $$\mathfrak{C}$$ is a Reedy category, then the cofibrations of the Reedy model category on $$\mathfrak{M}^{\mathfrak{C}}$$ are monomorphisms.

I think we can use this to prove the following: If we've got a Reedy model category $$\mathfrak{A}$$ and a category $$\mathcal{B}$$ in which all cofibrations are monomorphisms and weak equivalences are pointwise (i.e. just a rough way to say that it's equipped with an injective model structure) then in the category $$\text{Psh}(\mathfrak{A},\text{Psh}(\mathcal{B},\textit{Ssets}))$$ all Reedy cofibrations are monomorphisms.

Is there a way to have the converse work for some types of Reedy model categories? (By converse I mean all monomorphisms being Reedy cofibrations)

If I'm not mistaken, this will imply some sort of relation between model structures.

Edit: Big thanks to Simon for pointing out the vague, unexplicit nature of the question.

• I'm not the person who downvoted, but I find that what you are asking is not very clear. The paragraph starting with "I think" is very confusing. Do you mean that $\mathfrak{U}$ is a Reedy category ? and what do you mean by "$\mathcal{B}$ is a category where cofibrations are monomorphisms...", are you talking about the injective model structure on $Psh(\mathcal{B},Ssets)$ ? also it is not clear a converse to what you are asking ? to the corollary, to the statement in the above mentioned paragraph ? maybe you should state more explicit what kind of statement you would like. Commented Jun 7, 2022 at 14:39
• And the title of the question should give a hint to the topic, something like Cofibrations of Reedy model categories would be much more informative. Commented Jun 7, 2022 at 14:53
• Thanks a lot @SimonHenry for the comment, I've tried to be a bit more explicit now. Commented Jun 7, 2022 at 16:41
• Thanks Jochen for the advice, I've tried to modify the title accordingly. Commented Jun 7, 2022 at 16:42
• Somewhat related Commented Jul 3, 2022 at 17:22

This sort of things isn't true for a general Reedy category, but for an elegant one $$R$$ (see the link for the definition) if $$\mathcal{E}$$ is a Grothendieck topos (for e.g. simplicial presheaves on something) in which the cofibration are the monomorphisms, then in $$\mathcal{E}^{R^{op}}$$ the Reedy cofibration are exactly the monomorphism. This is explained on the nLab page linked and you'll also find reference about this there.