If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial sets. These are called the injective and projective model structures, and in both the weak equivalences are the levelwise weak equivalences, i.e. those natural transformations $X \to Y$ such that $X(c) \to Y(c)$ is a weak equivalence for all $c \in C$.
The cofibrations in the injective model structure are the levelwise cofibrations. The injective fibrations are then defined as those maps which have the left-lifting property with respect to all cofibrations which are also weak equivalences. The injective fibrant objects are those simplicial presheaves in which the map to the terminal presheaf is an injective fibration.
My Question: Is there a better characterization of the fibrant objects? If I have an object are there any shortcuts which I can use to test if it is fibrant? An obvious necessary condition is that each simplicial set $X(c)$ must be fibrant (i.e. a Kan complex), but I want sufficient conditions. I'm willing to put restrictions on C.
Dually, we can look at the projective model structure where the fibrations are the levelwise fibrations, and cofibrations have the extension property with respect to these. I'd also be interested in knowing about the cofibrant objects in this model structure, although for now I'm more interested in the injective model structure.
Best Answer
In the introduction to his paper "Flasque Model Structures for Presheaves" (in fact simplicial presheaves) Isaksen states on the top of page 2 that his model structure has a nice characterisation of fibrant objects and that "This is entirely unlike the injective model structures, where there is no explicit description of the fibrant objects". This would answer your question. It might be my ignorance, but I think there is no justification for Isaksen quoted statement except that no characterisation is known as yet.