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.

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 right-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.

If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category is 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 of 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.

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.