on dg-algebras/on dg-coalgebras and on on cosimplicial rings (related by monoidal Dold-Kan correspondence)
The model category structures on functor categories are models for (∞,1)-categories of (∞,1)-functors.
For a model category and any small category there are two “obvious” ways to put a model category structures on the functor category , called the projective and the injective model structure. For completely general , neither one need exist. The projective model structure exists as long as is cofibrantly generated, while the injective model structure exists as long as is combinatorial.
A related kind of model structure is the Reedy model structure on functor categories, which applies for any model category , but requires to be a very special sort of category. See the link for further information.
In the special case that SSet is the standard model category of simplicial sets the projective and injective model structure on the functor categories are described in more detail at global model structure on simplicial presheaves.
For a combinatorial model category (or, in the projective case, merely cofibrantly generated) and a small category there exists the following two (combinatorial) model category structures on the functor category :
the projective structure : weak equivalences and fibrations are the natural transformations that are objectwise such morphisms in .
the injective structure : weak equivalences and cofibrations are the natural transformations that are objectwise such morphisms in .
More generally, if is in addition a simplicial model category and a smooth sSet-enriched category, then the sSet-enriched functor category, also denoted , carries the above two model strutures.
For a combinatorial model category and a small category the projective and injective structures and
are indeed model category structures;
are themselves combinatorial model categories;
are right or left proper model categories of is right or left proper, respectively.
are simplicial model categories if is a simplicial model category, with respect to the sSet-enrichment for which the sSet-tensoring is objectwise that of .
The cofibrations in are generated from (i.e. are the weakly saturated class of morphisms defined by) the morphisms of the form
for all and a generating cofibration in . Here the dot denotes the tensoring of over sets, i.e. is the functor that sends to the coproduct of copies of .
In particular, every cofibration if is in particular a cofibration in . Similarly, every fibration in is in particular a fibration in
So the identity functors
form a Quillen equivalence (with being the right Quillen functor).
The functor model structures depend Quillen-functorially on their codomain, in that for
a Quillen adjunction between combinatorial model categories, composition induces Quillen adjunctions
and
If is sSet-enriched, then so is .
The Quillen-functoriality on the domain is more asymmetric: for a morphism of small categories, and , we have Quillen adjunctions
and
In the -enriched case, if is an equivalence in the model structure on sSet-categories, then these two Quillen adjunctions are both Quillen equivalences.
For the unenriched case this is HTT, prop A.2.8.2 and the following remarks. The enriched case is HTT, prop. A.3.3.2 and the remarks following that.
For a combinatorial simplicial model category, the (∞,1)-category presented by and under the above assumptions is the (∞,1)-category of (∞,1)-functors from the ordinary category to the -category presented by .
See (∞,1)-category of (∞,1)-functors for details.
Often functors are thought of as diagrams in the model category , and one is interested in obtaining their homotopy limit or homotopy colimit or, generally, for any functor, their left and right homotopy Kan extension.
These are the left and right derived functors and of
and
respectively.
For more on this see homotopy Kan extension. For the case that this reduces to homotopy limit and homotopy colimit.
The plain situation is the topic of section A.2.8 of
The enriched situation is section A.3.3 there.