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I equip the category of presheaves $[\mathcal{D}^{op},\text{Gpd}]$ with the injective model structure ($\mathcal{D}$ is just any small category). In this structure, weak equivalences and cofibrations are componentwise. One can notice that fibrations are in particular componentwise fibrations (but the converse is not true). Are there sufficient conditions on a componentwise fibration to be a (injective) fibration ?

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    $\begingroup$ I doubt there are any useful conditions. For instance, notice that the injective model structure on $[\mathbf{\Delta}^\mathrm{op}, \mathbf{sSet}]$ is the Reedy model structure, i.e. the injective fibrations are the Reedy fibrations. $\endgroup$ – Zhen Lin Feb 6 '14 at 19:09
  • $\begingroup$ Yes, it's probably not sensible to ask this level of generality, so I edited my question with $\mathcal{C} = \text{Gpd}$. $\endgroup$ – user123619 Feb 6 '14 at 22:49

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