on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
The model category structures on functor categories are models for (∞,1)-categories of (∞,1)-functors.
For $C$ a model category and $D$ any small category there are two “obvious” ways to put a model category structure on the functor category $[D,C]$, called the projective and the injective model structures. 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/generalized Reedy model structure on functor categories, which applies for any model category $C$, but requires $D$ to be a very special sort of category, namely a Reedy category/generalized Reedy category.
In the special case that $C =$ sSet is the classical model structure on simplicial sets the projective and injective model structure on the functor categories $[D,SSet]$ are described in more detail at global model structure on simplicial presheaves.
For $C$ a combinatorial model category or, in the projective case, just a cofibrantly generated model category, and $D$ a small category there exist the following two (combinatorial) model category structures on the functor category $[D,C]$:
the projective structure $[D,C]_{proj}$: weak equivalences and fibrations are the natural transformations that are objectwise such morphisms in $C$.
the injective structure $[D,C]_{inj}$: weak equivalences and cofibrations are the natural transformations that are objectwise such morphisms in $C$.
More generally, if $C$ is in addition a simplicial model category and $D$ a smooth sSet-enriched category, then the sSet-enriched functor category, also denoted $[D,C]$, carries the above two model strutures.
In all of the following, let $\mathbf{S}$ be an excellent model category. The standard example is the model structure on simplicial sets, $sSet_{Quillen}$.
Let $D$ (and $D_1$, $D_2$, …) be a combinatorial $\mathbf{S}$-enriched model category.
Moreover, $C$ in the following is assumed to be either an ordinary small category, or, more generally, it is a small $\mathbf{S}$-enriched category.
If $\mathbf{S} =$ sSet${}_{Quillen}$ and $C$ is an ordinary small category, then then model structures discussed here are instances of the model structure on simplicial presheaves. If $C$ is itself $sSet$-enriched, then they are instances of the model structure on sSet-enriched presheaves.
The projective and injective structures $[C,D]_{proj}$ and $[C,D]_{inj}$, def. 1
are indeed model category structures;
are themselves combinatorial model categories;
are right or left proper model categories if $C$ is right or left proper, respectively.
are $\mathbf{S}$-enriched model categories (e.g.simplicial model categories) with respect to the $\mathbf{S}$-enrichment for which the $\mathbf{S}$-tensoring is objectwise that of $C$.
The existence of the unenriched model structure apears as HTT, prop. A.2.8.2 The enriched case is HTT, prop. A.3.3.2 and the remarks following that. The statement about properness appears as HTT, remark A.2.8.4.
Let $C$ be an ordinary small category.
The cofibrations in $[C, A]_{proj}$ are generated from (i.e. are the weakly saturated class of morphisms defined by) the morphisms of the form
for all $c \in C$ and $i : a \to b$ a generating cofibration in $A$. Here the dot denotes the tensoring of $A$ over sets, i.e. $C(c,-)\cdot a$ is the functor that sends $c' \in C$ to the coproduct $\coprod_{C(c,c')} A$ of $|C(c,c')|$ copies of $A$.
In particular, every cofibration if $[C,A]_{proj}$ is in particular a cofibration in $[C,A]_{inj}$. Similarly, every fibration in $[C,A]_{inj}$ is in particular a fibration in $[C,A]_{proj}$
This is argued in the beginning of the proof of HTT, lemma A.2.8.3.
For $Top$-enriched functors, this is (Piacenza 91, section 5). For details see at classical model structure on topological spaces – Model structure on enriched functors.
form a Quillen equivalence (with $Id : [D,C]_{proj} \to [D,C]_{inj}$ being the left Quillen functor).
If $D$ is a Reedy category this factors through the Reedy model structure
The functor model structures depend Quillen-functorially on their codomain, in that for
a $\mathbf{S}$-enriched Quillen adjunction between combinatorial $\mathbf{S}$-enriched model categories, postcomposition induces $\mathbf{S}$-enriched Quillen adjunctions
and
Moreover, if $(L \dashv R)$ is a Quillen equivalence, then so is $([C,L] \dashv [C,R])$.
For the case that $C$ is a small category this is (Lurie, remark A.2.8.6), for the enriched case this is (Lurie, prop. A.3.3.6).
The Quillen-functoriality on the domain is more asymmetric.
For $p : C_1 \to C_2$ a functor between small categories or an $\mathbf{S}$-enriched functor between $\mathbf{S}$-enriched categories, let
be the adjoint triple where $p^*$ is precomposition with $p$ and where $p_!$ and $p_*$ are left and right Kan extension along $p$, respectively.
Then we have Quillen adjunctions
and
For $C$ not enriched this appears as (Lurie, prop. A.2.8.7), for the enriched case it appears as (Lurie, prop. A.3.3.7).
In the $sSet$-enriched case, if $p : D_1 \to D_2$ is an weak equivalence in the model structure on sSet-categories, then these two Quillen adjunctions are both Quillen equivalences.
For $C$ a combinatorial simplicial model category, the (∞,1)-category presented by $[D,C]_{proj}$ and $[D,C]_{inj}$ under the above assumptions is the (∞,1)-category of (∞,1)-functors $Func(D,C^\circ)$ from the ordinary category $D$ to the $(\infty,1)$-category presented by $C$.
See (∞,1)-category of (∞,1)-functors for details.
Often functors $D \to C$ are thought of as diagrams in the model category $C$, and one is interested in obtaining their homotopy limit or homotopy colimit or, generally, for $p : D \to D'$ any functor, their left and right homotopy Kan extension.
These are the left and right derived functors $HoLan := \mathbb{L} p_1$ and $HoRan := \mathbb{R} p_*$ of
and
respectively.
For more on this see homotopy Kan extension. For the case that $D' = *$ this reduces to homotopy limit and homotopy colimit.
Examples of cofibrant objects in the projective model structure are discussed at
The projective model structure on $Top_{Quillen}$-enriched functors is discussed in
Robert Piacenza section 5 of Homotopy theory of diagrams and CW-complexes over a category, Can. J. Math. Vol 43 (4), 1991 (pdf)
also chapter VI of Peter May et al., Equivariant homotopy and cohomology theory, 1996 (pdf)
See also
General review and discussion includes
Philip Hirschhorn, section 11.6 in Model categories and their localizations, 2003 (pdf)
Jacob Lurie, sections A.2.8 (unenriched) and section A.3.3 (enriched) of Higher Topos Theory, 2009
The projective model structure for functors to sSet on a large domain is discussed in
See also