nLab model structure on functors



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks



The model category structures on functor categories are models for (∞,1)-categories of (∞,1)-functors.

For CC a model category and DD any small category there are two “obvious” ways to put a model category structure on the functor category [D,C][D,C], called the projective and the injective model structures. For completely general CC, neither one need exist, but there are rather general conditions that ensure their existence. In particular, the projective model structure exists as long as CC is cofibrantly generated, while both model structures exist if CC is accessible (and in particular if it is combinatorial). In the case of enriched diagrams, additional cofibrancy-type conditions are required on DD.

A related kind of model structure is the Reedy model structure/generalized Reedy model structure on functor categories, which applies for any model category CC, but requires DD to be a very special sort of category, namely a Reedy category/generalized Reedy category.

In the special case that C=C = sSet is the classical model structure on simplicial sets the projective and injective model structure on the functor categories [D,SSet][D,SSet] are described in more detail at global model structure on simplicial presheaves and model structure on sSet-enriched presheaves.


Let S\mathbf{S} be a symmetric monoidal category, let CC be an S\mathbf{S}-model category that is an S\mathbf{S}-enriched category, and let DD be a small S\mathbf{S}-enriched category. Usually we have either S=Set\mathbf{S}=Set or else S\mathbf{S} is a monoidal model category and CC an S\mathbf{S}-enriched model category.

Let [D,C][D,C] denote the enriched functor category, whose objects are S\mathbf{S}-enriched functors DCD\longrightarrow C.


We define the following classes of maps in [D,C][D,C]:

  • the projective weak equivalences and projective fibrations are the natural transformations that are objectwise such morphisms in CC.
  • the injective weak equivalences and injective cofibrations are the natural transformations that are objectwise such morphisms in CC.

If either of these choices defines a model structure on [D,C][D,C], we call it the projective model structure [D,C] proj[D,C]_{proj} or injective model structure [D,C] inj[D,C]_{inj} respectively. Of course, the projective cofibrations and injective fibrations can then be characterized by lifting properties.


Projective case

The projective model structure can be regarded as a right-transferred model structure. This yields the following basic result on its existence.


(existence and cofibrant generation of projective structure on enriched fuinctors)

where CC carries the structure of

  1. a cofibrantly generated model category,

  2. which admits copowers by the hom-objects D(x,y)SD(x,y)\in \mathbf{S}, which preserve acyclic cofibrations.

    (This is the case for instance if S=\mathbf{S}= Set, or if S\mathbf{S} is a monoidal model category, CC is an S \mathbf{S} -model category, and the hom-objects D(x,y)D(x,y) are cofibrant in S\mathbf{S}.)

Then the projective model structure [D,C] proj[D,C]_{proj} on the enriched functor category exists, and is again cofibrantly generated.

(For S=\mathbf{S} = Sets this is Hirschhorn 2002, Thm. 11.6.1.).

Assuming the existence of such copowers, for any xob(D)x\in ob(D) the “evaluation at xx” functor ev x:[D,C]Cev_x \colon [D,C]\longrightarrow C has a left adjoint F xF_x sending ACA\in C to the functor yD(x,y)Ay\mapsto D(x,y)\odot A, where \odot denotes the copower. Now if II and JJ are generating sets of cofibrations and trivial cofibrations for CC, let I DI^D be the set of maps F x(i)F_x(i) in [D,C][D,C], for all iIi\in I and xob(D)x\in ob(D), and similarly for JJ. Then the projective fibrations and trivial fibrations are characterized by having the right lifting property with respect to J DJ^D and I DI^D respectively, while both I DI^D and J DJ^D permit the small object argument since II and JJ do and colimits in [D,C][D,C] are pointwise. Since the trivial fibrations in [D,C][D,C] clearly coincide with the fibrations that are weak equivalences, it remains only to show that all J DJ^D-cell complexes are weak equivalences. But a J DJ^D-cell complex is objectwise a cell complex built from cells D(x,y)jD(x,y)\odot j for maps jJj\in J, and the assumption ensures that these are trivial cofibrations in CC, 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 CC is a locally presentable 2-category with its 2-trivial model structure and DD is a small 2-category, then the projective model structure on [D,C][D,C] exists.


This follows from the result of Lack on transferred model structures for algebras over 2-monads, since [D,C][D,C] is the category of algebras for an accessible 2-monad on C ob(D)C^{ob(D)}.

Note that CC 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.

Accessible case

In the case when CC 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).




  1. If copowers by the hom-objects D(x,y)D(x,y) preserve acyclic cofibrations, then the projective model structure on [D,C][D,C] exists and is accessible.

  2. If copowers by the hom-objects D(x,y)D(x,y) preserve cofibrations, then the injective model structure on [D,C][D,C] exists and is accessible

Combinatorial case

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 (by this Prop.) 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 & Rosický 2014 to essentially the following:


Let S\mathbf{S} be a locally presentable cosmos, CC an S\mathbf{S}-cocomplete locally S\mathbf{S}-presentable S\mathbf{S}-enriched category that is a combinatorial model category, and DD a small S\mathbf{S}-category. Then:

  1. If copowers by the hom-objects D(x,y)D(x,y) preserve trivial cofibrations, then the projective model structure on [D,C][D,C] exists and is combinatorial.

  2. If copowers by the hom-objects D(x,y)D(x,y) preserve cofibrations, then the injective model structure on [D,C][D,C] exists and is combinatorial.


It suffices to construct the factorizations, which follows from Makkai & Rosický 2014, Remark 3.8 about left-lifting combinatorial weak factorization systems.




The projective and injective structures [D,C] proj[D,C]_{proj} and [D,C] inj[D,C]_{inj}, def. , are (insofar as they exist):

The statement about properness appears as HTT, remark A.2.8.4.


For CC a combinatorial simplicial model category, the (∞,1)-category presented by [D,C] proj[D,C]_{proj} and [D,C] inj[D,C]_{inj} under the above assumptions is the (∞,1)-category of (∞,1)-functors Func(D,C )Func(D,C^\circ) from the ordinary category DD to the (,1)(\infty,1)-category presented by CC.

See at (∞,1)-category of (∞,1)-functors for more.

Relation to other model structures


If copowers by the hom-objects of DD preserve trivial cofibrations, then every every fibration in [D,C] inj[D,C]_{inj} is in particular a fibration in [D,C] proj[D,C]_{proj}. Similarly, if powers by the hom-objects of DD preserve trivial fibrations, then every cofibration in [D,C] proj[D,C]_{proj} is in particular a cofibration in [D,C] inj[D,C]_{inj}. The hypotheses are satisfied if DD is unenriched, or in the monoidal model category case if the hom-objects of DD are cofibrant.

This is argued in the beginning of the proof of HTT, lemma A.2.8.3. For TopTop-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:ABi:A\longrightarrow B is a trivial cofibration in CC and xob(D)x\in ob(D), then the first assumption implies that F x(i):F x(A)F x(B)F_x(i) : F_x(A) \longrightarrow F_x(B), for F x(A)(y)=D(x,y)AF_x(A) (y) = D(x,y) \odot A the left adjoint of ev x:[D,C]Cev_x : [D,C] \longrightarrow C, is a trivial cofibration in [D,C] inj[D,C]_{inj}. Thus, any fibration pp in [D,C] inj[D,C]_{inj} has the right lifting property with respect to it, which is to say that ev x(p)ev_x(p) has the right lifting property with respect to ii. Since this is true for any ii, each ev x(p)ev_x(p) is a fibration, i.e. pp is a fibration in [D,C] inj[D,C]_{inj}. The other half is dual.


The identity functors

[D,C] injIdId[D,C] proj [D,C]_{inj} \stackrel{\overset{Id}{\longleftarrow}}{\underset{Id}{\longrightarrow}} [D,C]_{proj}

form a Quillen equivalence (with Id:[D,C] proj[D,C] injId : [D,C]_{proj} \longrightarrow [D,C]_{inj} being the left Quillen functor).

If DD is a Reedy category this factors through the Reedy model structure

[D,C] injIdId[D,C] ReedyIdId[D,C] proj [D,C]_{inj} \stackrel{\overset{Id}{\longleftarrow}}{\underset{Id}{\longrightarrow}} [D,C]_{Reedy} \stackrel{\overset{Id}{\longleftarrow}}{\underset{Id}{\longrightarrow}} [D,C]_{proj}

Functoriality in domain and codomain


The functor model structures depend Quillen-functorially on their codomain, in that for

D 1RLD 2 D_1 \stackrel {\overset{L}{\longleftarrow}} {\underset{R}{\longrightarrow}} D_2

an S\mathbf{S}-enriched Quillen adjunction between combinatorial S\mathbf{S}-enriched model categories, postcomposition induces S\mathbf{S}-enriched Quillen adjunctions

[C,D 1] proj[C,R][C,L][C,D 2] proj [C,D_1]_{proj} \stackrel {\overset{[C,L]}{\longleftarrow}} {\underset{[C,R]}{\longrightarrow}} [C,D_2]_{proj}


[C,D 1] inj[C,R][C,L][C,D 2] inj. [C,D_1]_{inj} \stackrel {\overset{[C,L]}{\longleftarrow}} {\underset{[C,R]}{\longrightarrow}} [C,D_2]_{inj} \,.

Moreover, if (LR)(L \dashv R) is a Quillen equivalence, then so is ([C,L][C,R])([C,L] \dashv [C,R]).

For the case that CC 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.


(Quillen functoriality in the domain category)
For p:C 1C 2p \colon C_1 \longrightarrow C_2 a functor between small categories or an S\mathbf{S}-enriched functor between S\mathbf{S}-enriched categories, let

(p !p *p *):[C 2,D]p *p *p ![C 1,D] (p_! \dashv p^* \dashv p_*) \;\colon\; [C_2,D] \stackrel{\overset{p_!}{\longleftarrow}}{\stackrel{\overset{p^*}{\longrightarrow}}{\underset{p_*}{\longleftarrow}}} [C_1,D]

be the adjoint triple where p *p^* is precomposition with pp and where p !p_! and p *p_* are left and right Kan extension along pp, respectively.

Then we have Quillen adjunctions

(p !p *):[C 1,D] projp *p ![C 2,D] proj (p_! \dashv p^*) \;\colon\; [C_1,D]_{proj} \stackrel{\overset{p_!}{\longrightarrow}}{\underset{p^*}{\longleftarrow}} [C_2,D]_{proj}


(p *p *):[C 1,D] injp *p *[C 2,D] inj. (p^* \dashv p_*) \;\colon\; [C_1,D]_{inj} \stackrel{\overset{p^*}{\longleftarrow}}{\underset{p_*}{\longrightarrow}} [C_2,D]_{inj} \,.

For CC 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 sSetsSet-enriched case, if p:D 1D 2p : D_1 \longrightarrow D_2 is a weak equivalence in the model structure on sSet-categories, then these two Quillen adjunctions are both Quillen equivalences.

Relation to homotopy Kan extensions/limits/colimits

Often functors DCD \longrightarrow C are thought of as diagrams in the model category CC, and one is interested in obtaining their homotopy limit or homotopy colimit or, generally, for p:DDp : D \longrightarrow D' any functor, their left and right homotopy Kan extension.

These are the left and right derived functors HoLan:=𝕃p 1HoLan := \mathbb{L} p_1 and HoRan:=p *HoRan := \mathbb{R} p_* of

[D,C] projp ![D,C] proj [D,C]_{proj} \stackrel{p_!}{\longrightarrow} [D',C]_{proj}


[D,C] injp *[D,C] inj [D,C]_{inj} \stackrel{p_*}{\longrightarrow} [D',C]_{inj}


For more on this see homotopy Kan extension. For the case that D=*D' = * this reduces to homotopy limit and homotopy colimit.

Relation to \infty-functor categories



then both the projective and the injective model structure on the sSet-enriched functor category sFunc(D,C)sFunc(\mathbf{D}, \mathbf{C}) (which exist by the above discussion) present the corresponding \infty -category of \infty -functors. Concretely:

The homotopy coherent nerve of the full sSet-enriched subcategory of the functor model category on its bifibrant objects, (sFunc(D,C)) \big(\mathbf{sFunc}(\mathbf{D}, \mathbf{C})\big)^\circ, is equivalent to the \infty -functor quasi-category between the homotopy coherent nerves

N(sFunc(D,C) )qCatFunc (N(D),N(C )). N \big( \mathbf{sFunc}(\mathbf{D}, \mathbf{C})^\circ \big) \;\; \underset{qCat}{\simeq} \;\; Func_\infty \Big( N(\mathbf{D}) ,\, N\big( \mathbf{C}^\circ \big) \Big) \,.

This is due to Lurie (2009), Prop. See also the discussion here at \infty -category of \infty -functors.


Examples of cofibrant objects in the projective model structure are discussed at projectively cofibrant diagram.


(model structure on simplicial presheaves)
Model structures on sSet-valued functors, hence on simplicial presheaves, play a central role in the theory of combinatorial model categories and specifically in model topos-theory presenting \infty -toposes. See at model structure on simplicial presheaves for more.



(Borel model structure on simplicial group actions)
For 𝒢Grp(sSet)\mathcal{G} \,\in\, Grp(sSet) a simplicial group with sSet-enriched delooping groupoid denoted B𝒢sSet-Grpd\mathbf{B}\mathcal{G} \in sSet\text{-}Grpd, an sSet-enriched functor B𝒢sSet\mathbf{B}\mathcal{G} \longrightarrow sSet is equivalently a simplicial group action of 𝒢\mathcal{G}.

Under this identification, the projective model strcuture on simplicial functors (Prop. ) is equivalently the Borel model structure on simplicial group actions, a context of Borel-equivariant homotopy theory:

sFunc(B𝒢,sSet) proj=𝒢Act(sSet) Borel. sFunc\big( \mathbf{B}\mathcal{G} ,\, sSet \big)_{proj} \;\; = \;\; \mathcal{G}Act(sSet)_{Borel} \,.

More generally, for 𝒳sSet-Grpd\mathcal{X} \in sSet\text{-}Grpd an sSet-enriched groupoid (Dwyer-Kan simplicial groupoid) with a single connected component π 0(𝒳){[x]}\pi_0(\mathcal{X}) \simeq \{[x]\}, so that the inclusion

ι:B(𝒳(x,x))𝒳 \iota \,\colon\, \mathbf{B}(\mathcal{X}(x,x)) \xhookrightarrow{\phantom{---}} \mathcal{X}

is an sSet-enriched adjoint equivalence (see discussion there) the projective model structure on simplicial functors from Prop. is transferred under the induced adjoint equivalence of sSet-enriched functor categories

sFunc(𝒳,sSet) proj ι *ι !sFunc(B(𝒳(x,x)),sSet) proj=(𝒳(x,x))Act(sSet) Borel sFunc\big( \mathcal{X} ,\, sSet \big)_{proj} \underoverset {\underset{\iota^\ast}{\longrightarrow}} {\overset{\iota_!}{\longleftarrow}} {\;\; \bot_{\simeq} \;\;} sFunc\Big( \mathbf{B}\big(\mathcal{X}(x,x)\big) ,\, sSet \Big)_{proj} \;=\; \big(\mathcal{X}(x,x)\big) Act(sSet)_{Borel}

By this example it follows that morphisms in all three classes (W,Fib,Cof)(\mathrm{W}, Fib, Cof) in sFunc(𝒳,sSet) projsFunc(\mathcal{X}, \, sSet)_{proj} are those which restrict on xObj(𝒳)x \in Obj(\mathcal{X}) to the respective class in (𝒳(x,x))Act(sSet) Borel\big(\mathcal{X}(x,x)\big) Act(sSet)_{Borel}.

It follows that for 𝒳sSet-Grpd\mathcal{X} \,\in\, sSet\text{-}Grpd a simplicial groupoid with any set π 0(𝒳)\pi_0(\mathcal{X}) of connected components, the projective model structure of simplicial functors over it is the product model structure of the Borel model structures of simplicial group actions, one for each connected component:

sFunc(𝒳,sSet) proj ι *ι !iπ 0(𝒳)sFunc(B(𝒳(x i,x i)),sSet) proj=iπ 0(𝒳)(𝒳(x i,x i))Act(sSet) Borel. sFunc\big( \mathcal{X} ,\, sSet \big)_{proj} \underoverset {\underset{\iota^\ast}{\longrightarrow}} {\overset{\iota_!}{\longleftarrow}} {\;\; \bot_{\simeq} \;\;} \underset{i \in \pi_0(\mathcal{X})}{\prod} sFunc\Big( \mathbf{B}\big(\mathcal{X}(x_i,x_i)\big) ,\, sSet \Big)_{proj} \;=\; \underset{i \in \pi_0(\mathcal{X})}{\prod} \big(\mathcal{X}(x_i,x_i)\big) Act(sSet)_{Borel} \,.


The projective model structure on Top QuillenTop_{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

  • Alex Heller, Homotopy in functor categories, Transactions of the AMS, vol 272, Number 1, July 1982 (JSTOR)

Textbook account of the projective model structure

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

It was generalized to enriched diagrams in

  • Lyne Moser, Injective and Projective Model Structures on Enriched Diagram Categories, arXiv:1710.11388

The more general result above on combinatoriality of injective model structures follows from Remark 3.8 of

See also

Last revised on November 5, 2023 at 13:22:27. See the history of this page for a list of all contributions to it.