model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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, but there are rather general conditions that ensure their existence. In particular, the projective model structure exists as long as $C$ is cofibrantly generated, while both model structures exist if $C$ is accessible (and in particular if it is combinatorial). In the case of enriched diagrams, additional cofibrancy-type conditions are required on $D$.
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 and model structure on sSet-enriched presheaves.
Let $\mathbf{S}$ be a symmetric monoidal category, let $C$ be an $\mathbf{S}$-model category that is an $\mathbf{S}$-enriched category, and let $D$ be a small $\mathbf{S}$-enriched category. Usually we have either $\mathbf{S}=Set$ or else $\mathbf{S}$ is a monoidal model category and $C$ an $\mathbf{S}$-enriched model category.
Let $[D,C]$ denote the enriched functor category, whose objects are $\mathbf{S}$-enriched functors $D\to C$.
We define the following classes of maps in $[D,C]$:
If either of these choices defines a model structure on $[D,C]$, we call it the projective model structure $[D,C]_{proj}$ or injective model structure $[D,C]_{inj}$ respectively. Of course, the projective cofibrations and injective fibrations can then be characterized by lifting properties.
The projective model structure can be regarded as a right-transferred model structure. This yields the following basic result on its existence.
Suppose that
Then the projective model structure $[D,C]_{proj}$ exists, and is again cofibrantly generated.
Assuming the existence of such copowers, for any $x\in ob(D)$ the “evaluation at $x$” functor $ev_x : [D,C]\to C$ has a left adjoint $F_x$ sending $A\in C$ to the functor $y\mapsto D(x,y)\odot A$, where $\odot$ denotes the copower. Now if $I$ and $J$ are generating sets of cofibrations and trivial cofibrations for $C$, let $I^D$ be the set of maps $F_x(i)$ in $[D,C]$, for all $i\in I$ and $x\in ob(D)$, and similarly for $J$. Then the projective fibrations and trivial fibrations are characterized by having the right lifting property with respect to $J^D$ and $I^D$ respectively, while both $I^D$ and $J^D$ permit the small object argument since $I$ and $J$ do and colimits in $[D,C]$ are pointwise. Since the trivial fibrations in $[D,C]$ clearly coincide with the fibrations that are weak equivalences, it remains only to show that all $J^D$-cell complexes are weak equivalences. But a $J^D$-cell complex is objectwise a cell complex built from cells $D(x,y)\odot j$ for maps $j\in J$, and the assumption ensures that these are trivial cofibrations in $C$, hence so is any cell complex built from them.
There do exist projective model structures that do not fall under this theorem, however, such as the following.
If $C$ is a locally presentable 2-category with its 2-trivial model structure and $D$ is a small 2-category, then the projective model structure on $[D,C]$ exists.
This follows from the result of Lack on transferred model structures for algebras over 2-monads, since $[D,C]$ is the category of algebras for an accessible 2-monad on $C^{ob(D)}$.
Note that $C$ need not be cofibrantly generated (and the 2-trivial model structure often fails to be cofibrantly generated), so the generality of this result is not entirely included in the previous one.
In the case when $C$ is an accessible model category, i.e. it is a locally presentable category and its constituent weak factorization systems have accessible realizations as functorial factorizations, we have the following general result from Moser (the unenriched case appears in HKRS15 and GKR18).
Let $\mathbf{S}$ be a locally presentable base?, $C$ an $\mathbf{S}$-cocomplete locally $\mathbf{S}$-presentable $\mathbf{S}$-enriched category that is an accessible model category, and $D$ a small $\mathbf{S}$-category. Then:
Every combinatorial model category (i.e. locally presentable and cofibrantly generated) is accessible, so Theorem shows that both model structures exist, and Theorem shows that the projective model structure is cofibrantly generated, hence also combinatorial. In fact the injective model structure is also combinatorial, although the proof is much more involved, because there is no explicit description of the generating cofibrations and acyclic cofibrations; they have to be produced by a cardinality argument. This was first proven by in HTT, prop. A.2.8.2 and A.3.3.2 under strong assumptions on the enriching category (in particular that all objects are cofibrant), and later generalized by Makkai and Rosicky to essentially the following:
Let $\mathbf{S}$ be a locally presentable base?, $C$ an $\mathbf{S}$-cocomplete locally $\mathbf{S}$-presentable $\mathbf{S}$-enriched category that is a combinatorial model category, and $D$ a small $\mathbf{S}$-category. Then:
It suffices to construct the factorizations, which follows from MR13, Remark 3.8 about left-lifting combinatorial weak factorization systems.
The projective and injective structures $[D,C]_{proj}$ and $[D,C]_{inj}$, def. , are (insofar as they exist):
right or left proper model categories if $C$ is right or left proper, respectively.
$\mathbf{S}$-enriched model categories if $C$ is an $\mathbf{S}$-model category.
The statement about properness appears as HTT, remark A.2.8.4.
If copowers by the hom-objects of $D$ preserve trivial cofibrations, then every every fibration in $[D,C]_{inj}$ is in particular a fibration in $[D,C]_{proj}$. Similarly, if powers by the hom-objects of $D$ preserve trivial fibrations, then every cofibration in $[D,C]_{proj}$ is in particular a cofibration in $[D,C]_{inj}$. The hypotheses are satisfied if $D$ is unenriched, or in the monoidal model category case if the hom-objects of $D$ are cofibrant.
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.
If $i:A\to B$ is a trivial cofibration in $C$ and $x\in ob(D)$, then the first assumption implies that $F_x(i) : F_x(A) \to F_x(B)$, for $F_x(A) (y) = D(x,y) \odot A$ the left adjoint of $ev_x : [D,C] \to C$, is a trivial cofibration in $[D,C]_{inj}$. Thus, any fibration $p$ in $[D,C]_{inj}$ has the right lifting property with respect to it, which is to say that $ev_x(p)$ has the right lifting property with respect to $i$. Since this is true for any $i$, each $ev_x(p)$ is a fibration, i.e. $p$ is a fibration in $[D,C]_{inj}$. The other half is dual.
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
The injective model structure for unenriched diagrams of simplicial sets was first constructed by
Probably the first general construction of injective model structures for enriched diagrams in combinatorial model categories was in
The projective model structure for functors to sSet on a large domain is discussed in
The case of diagrams in a 2-category is a special case of
The use of accessible model structures to construct both projective and injective model structures on unenriched diagrams was introduced in
Marzieh Bayeh, Kathryn Hess, Varvara Karpova, Magdalena Kędziorek, Emily Riehl, Brooke Shipley, Left-induced model structures and diagram categories (arXiv:1401.3651)
Kathryn Hess, Magdalena Kędziorek, Emily Riehl, Brooke Shipley, A necessary and sufficient condition for induced model structures (arXiv:1509.08154). This paper contains an error, corrected by:
Richard Garner, Magdalena Kedziorek, Emily Riehl, Lifting accessible model structures, arXiv:1802.09889
It was generalized to enriched diagrams in
The more general result above on combinatoriality of injective model structures follows from Remark 3.8 of
See also