nLab
global model structure on functors

Idea

For $C$ a model category and $D$ any small category there are two “obvious” ways to put a model category structures on the functor category $[D,C]$, called the projective and the injective model structure. For completely general $C$, neither one need exist. The projective model structure exists as long as $C$ is cofibrantly generated, while the injective model structure exists as long as $C$ is combinatorial.

A related kind of model structure is the Reedy model structure on functor categories, which applies for any model category $C$, but requires $D$ to be a very special sort of category. See the link for further information.

In the special case that $C=$ SSet is the standard model category of simplicial sets the projective and injective model structure on the functor categories $[D,\mathrm{SSet}]$ are described in more detail at global model structure on simplicial presheaves.

Definition

For $C$ a combinatorial model category (or, in the projective case, merely cofibrantly generated) and $D$ a small category there exists the following two (combinatorial) model category structures on the functor category $[D,C]$:

• the projective structure $[D,C{]}_{\mathrm{proj}}$: weak equivalences and fibrations are the natural transformations that are objectwise such morphisms in $C$. (The cofibrations are then defined by their left lifting property.)

• the injective structure $[D,C{]}_{\mathrm{inj}}$: weak equivalences and cofibrations are the natural transformations that are objectwise such morphisms in $C$. (The fibrations are then defined by their right lifting property.)

Proposition

For $C$ a combinatorial model category and $D$ a small category the projective and injective structures $[D,C{]}_{\mathrm{proj}}$ and $[D,C{]}_{\mathrm{inj}}$ are themselves combinatorial model categories.

Proof

This is for instance HTT, prop A.2.8.2

References

For instance proposition A.2.8.2 of

• Jacob Lurie, Higher Topos Theory