nLab model structure on functors

Idea

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.

Definition

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 $D\longrightarrow C$.

Definition

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

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

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.

Existence

Projective case

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

Theorem

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

• $\mathbf{S}$ a cosmos for enrichment,

• $D$, $C$ a pair of $\mathbf{S}$-enriched categories

where $C$ carries the structure of

1. a cofibrantly generated model category,

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

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

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

Proof

Assuming the existence of such copowers, for any $x\in ob(D)$ the “evaluation at $x$” functor $ev_x \colon [D,C]\longrightarrow C$ 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.

Theorem

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.

Proof

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.

Accessible case

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).

Theorem

Let

• $\mathbf{S}$ be a locally presentable cosmos

• $C$ an $\mathbf{S}$-cocomplete locally $\mathbf{S}$-presentable $\mathbf{S}$-enriched category that is an accessible model category,

• $D$ a small $\mathbf{S}$-category.

Then:

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

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

Theorem

Let $\mathbf{S}$ be a locally presentable cosmos, $C$ 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:

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

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

Proof

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

Properties

General

Proposition

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.

Proposition

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

Relation to other model structures

Proposition

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.

Proof

If $i:A\longrightarrow B$ is a trivial cofibration in $C$ and $x\in ob(D)$, then the first assumption implies that $F_x(i) : F_x(A) \longrightarrow F_x(B)$, for $F_x(A) (y) = D(x,y) \odot A$ the left adjoint of $ev_x : [D,C] \longrightarrow C$, 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.

Corollary

The identity functors

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

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

If $D$ is a Reedy category this factors through the Reedy model structure

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

Functoriality in domain and codomain

Proposition

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

$$D_1 \stackrel {\overset{L}{\longleftarrow}} {\underset{R}{\longrightarrow}} D_2$$

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

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

and

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

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.

Proposition

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

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

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

Then we have Quillen adjunctions

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

and

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

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].

Remark

In the $sSet$-enriched case, if $p : D_1 \longrightarrow D_2$ 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 $D \longrightarrow C$ 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 $p : D \longrightarrow D'$ 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

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

and

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

respectively.

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

Relation to $\infty$-functor categories

Proposition

For

• $\mathbf{C}$ a combinatorial simplicial model category

• $\mathbf{D}$ a small sSet-enriched category

then both the projective and the injective model structure on the sSet-enriched functor category $sFunc(\mathbf{D}, \mathbf{C})$ (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, $\big(\mathbf{sFunc}(\mathbf{D}, \mathbf{C})\big)^\circ$, is equivalent to the $\infty$-functor quasi-category between the homotopy coherent nerves

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

This is due to Lurie (2009), Prop. 4.2.4.4. See also the discussion here at $\infty$-category of $\infty$-functors.

Examples

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

Example

(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.

Specifically:

Example

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

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:

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

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

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

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

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

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

It follows that for $\mathcal{X} \,\in\, sSet\text{-}Grpd$ a simplicial groupoid with any set $\pi_0(\mathcal{X})$ 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:

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

Related concepts

• Reedy model structure, generalized Reedy model structure,

• model structure on sections

• lim^1 and Milnor sequences

• model structure on homotopical presheaves, model structure on simplicial presheaves

• model structure for n-excisive functors

References

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

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

General review and discussion includes

• Philip Hirschhorn, section 11.6 in Model categories and their localizations, 2003 (pdf)

The injective model structure for unenriched diagrams of simplicial sets was first constructed by

• Alex Heller, Homotopy theories

Probably the first general construction of injective model structures for enriched diagrams in combinatorial model categories was in

• 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

• Boris Chorny, William Dwyer, Homotopy theory of small diagrams over large categories, arXiv:math/0607117

The case of diagrams in a 2-category is a special case of

• Steve Lack, Homotopy-theoretic aspects of 2-monads, arXiv

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

• 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

• M. Makkai, J. Rosický, Cellular categories, J. Pure Appl. Alg. 218 (2014) 1652-1664 [arXiv:1304.7572, doi:10.1016/j.jpaa.2014.01.005]

See also

• David White, Modified projective model structure (MO comment)