# Model structure on stacks

does anyone know if there is a model structure on stacks where the cofibrations are the monomorphisms ?
As far as I know usually to get a model structure on stacks one localizes a model structure on presheaves of groupoids or sheaves of groupoids or categories fibered in groupoids and this process doesn't change the cofibrations so my question may be reformulated as: is there a model structure on those categories above where the cofibrations are the monomorphisms ? I would be grateful for a reference if any.

Best

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I don't know why my "Hi" at the beginning is deleted ? –  Asymptotik Dec 16 '13 at 11:07
What about the Joyal–Jardine model structure on simplicial presheaves? The homotopy category thereof contains the homotopy category of stacks of groupoids, and I imagine a further Bousfield localisation will yield a model structure that is Quillen equivalent to the Joyal–Tierney model structure for strong stacks. –  Zhen Lin Dec 16 '13 at 11:32
Correct me if I'm wrong but in jardine's model structure on simplicial presheaves cofibrations are the objectwise cofibrations and so are exactly the monomorphisms (since monos in sSet are objectwise monos ,ie injective maps, in Sets and sSet has pullbacks so monos in simplicial presheaves are exactly objectwise monos) so it can help. –  Asymptotik Dec 16 '13 at 16:20
@Asymptotik maybe you want to impose a given class of weak equivalences, so as to avoid 'trivial' model structures. –  Fernando Muro Dec 16 '13 at 18:53
Yes, I agree Fernando. And for my purpose (limits preserve cofibrations) that cofibrations are monomorphisms is just a sufficient condition but not needed. –  Asymptotik Dec 17 '13 at 16:50