If $M$ is a combinatorial model category and $C$ a small category, then the category $M^C$ admits an injective model structure in which the cofibrations and weak equivalences are levelwise. I would like to have an explicit description of the injective fibrations, but this seems to be a very hard problem in general. The only case in which I know the answer is when $C$ is a Reedy category and the injective model structure coincides with the Reedy model structure (which is the case for some Reedy categories, though not all of them). So let's start with the simplest possible nontrivial case that I can think of.

Let $M=\mathrm{Gpd}$ be the category of groupoids, with its canonical model structure in which the weak equivalences are the equivalences of groupoids, the cofibrations are the functors that are injective on objects, and the fibrations have the isomorphism-lifting property. Is there any small category $C$ which is not a Reedy category, but for which you can give an explicit description of the fibrations in the injective model structure on $\mathrm{Gpd}^C$?

  • $\begingroup$ Can you think of candidates for a simple example of $C$? I assume you probably also don't want it to be a generalized Reedy category... $\endgroup$ – Dylan Wilson Mar 6 '13 at 14:34
  • $\begingroup$ Pick whatever $C$ you like! (Although yes, a generalized Reedy category would also not be very helpful...) $\endgroup$ – Mike Shulman Mar 7 '13 at 3:40

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