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 structures on the functor category $[D,C]$, called the projective and the injective model structure. 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 on functor categories, which applies for any model category $C$, but requires $D$ to be a very special sort of category. See the link for further information.

In the special case that $C=$ SSet is the standard model category of simplicial sets the projective and injective model structure on the functor categories $[D,\mathrm{SSet}]$ are described in more detail at global model structure on simplicial presheaves.

Definition

Definition

For $C$ a combinatorial model category (or, in the projective case, merely cofibrantly generated) and $D$ a small category there exists the following two (combinatorial) model category structures on the functor category $[D,C]$:

• the projective structure $[D,C{]}_{\mathrm{proj}}$: weak equivalences and fibrations are the natural transformations that are objectwise such morphisms in $C$.

• the injective structure $[D,C{]}_{\mathrm{inj}}$: 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.

Properties

Proposition

For $C$ a combinatorial model category and $D$ a small category the projective and injective structures $[D,C{]}_{\mathrm{proj}}$ and $[D,C{]}_{\mathrm{inj}}$

• are indeed model category structures;

• are themselves combinatorial model categories;

• are right or left proper model categories of $C$ is right or left proper, respectively.

• are simplicial model categories if $C$ is a simplicial model category, with respect to the sSet-enrichment for which the sSet-tensoring is objectwise that of $C$.

The cofibrations in $[C,A{]}_{\mathrm{proj}}$ are generated from (i.e. are the weakly saturated class of morphisms defined by) the morphisms of the form

$${\mathrm{Id}}_{C(c,-)}\cdot i:C(c,-)\cdot a\to C(c,-)\cdot b$$Id_{C(c,-)}\cdot i : C(c,-)\cdot a \to C(c,-) \cdot b

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 $c\prime \in C$ to the coproduct ${\coprod }_{C(c,c\prime )}A$ of $\mid C(c,c\prime )\mid$ copies of $A$.

In particular, every cofibration if $[C,A{]}_{\mathrm{proj}}$ is in particular a cofibration in $[C,A{]}_{\mathrm{inj}}$. Similarly, every fibration in $[C,A{]}_{\mathrm{inj}}$ is in particular a fibration in $[C,A{]}_{\mathrm{proj}}$

So the identity functors

$$[D,C{]}_{\mathrm{proj}}\stackrel{\stackrel{\mathrm{Id}}{\leftarrow }}{\underset{\mathrm{Id}}{\to }}[D,C{]}_{\mathrm{inj}}$$[D,C]_{proj} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [D,C]_{inj}

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

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

$$C_1\stackrel{\stackrel{L}{\leftarrow }}{\underset{R}{\to }}{C}_2$$C_1 \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} C_2

a Quillen adjunction between combinatorial model categories, composition induces Quillen adjunctions

$$[D,C_1{]}_{\mathrm{proj}}\stackrel{\stackrel{[D,L]}{\leftarrow }}{\underset{[D,R]}{\to }}[D,C_2{]}_{\mathrm{proj}}$$[D,C_1]_{proj} \stackrel{\overset{[D,L]}{\leftarrow}}{\underset{[D,R]}{\to}} [D,C_2]_{proj}

and

$$[D,C_1{]}_{\mathrm{inj}}\stackrel{\stackrel{[D,L]}{\leftarrow }}{\underset{[D,R]}{\to }}[D,C_2{]}_{\mathrm{inj}}.$$[D,C_1]_{inj} \stackrel{\overset{[D,L]}{\leftarrow}}{\underset{[D,R]}{\to}} [D,C_2]_{inj} \,.

If $(L⊣R)$ is sSet-enriched, then so is $([D,L]⊣[D,R])$.

The Quillen-functoriality on the domain is more asymmetric: for $p:D_1\to D_2$ a morphism of small categories, and $p^{*}=[p,C]:[D_2,C]\to [D_1,C]$, we have Quillen adjunctions

$$(p_{!}⊣p^{*}):[D_1,C{]}_{\mathrm{proj}}\stackrel{\stackrel{p_{!}}{\to }}{\underset{p^{*}}{\leftarrow }}[D_2,C{]}_{\mathrm{proj}}$$(p_! \dashv p^*) : [D_1,C]_{proj} \stackrel{\overset{p_!}{\to}}{\underset{p^*}{\leftarrow}} [D_2,C]_{proj}

and

$$(p^{*}⊣p_{*}):[D_1,C{]}_{\mathrm{inj}}\stackrel{\stackrel{p^{*}}{\leftarrow }}{\underset{p_{*}}{\to }}[D_2,C{]}_{\mathrm{inj}}.$$(p^* \dashv p_*) : [D_1,C]_{inj} \stackrel{\overset{p^*}{\leftarrow}}{\underset{p_*}{\to}} [D_2,C]_{inj} \,.

In the $\mathrm{sSet}$-enriched case, if $p:D_1\to D_2$ is an equivalence in the model structure on sSet-categories, then these two Quillen adjunctions are both Quillen equivalences.

Proof

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.

Proposition

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

Proof

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

Relation to homotopy Kan extensions/limits/colimits

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 $p:D\to D\prime$ any functor, their left and right homotopy Kan extension.

These are the left and right derived functors $\mathrm{HoLan}:=𝕃p_1$ and $\mathrm{HoRan}:=ℝp_{*}$ of

$$[D,C{]}_{\mathrm{proj}}\stackrel{p_{!}}{\to }[D\prime ,C{]}_{\mathrm{proj}}$$[D,C]_{proj} \stackrel{p_!}{\to} [D',C]_{proj}

and

$$[D,C{]}_{\mathrm{inj}}\stackrel{p_{*}}{\to }[D\prime ,C{]}_{\mathrm{inj}}$$[D,C]_{inj} \stackrel{p_*}{\to} [D',C]_{inj}

respectively.

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

References

The plain situation is the topic of section A.2.8 of

• Jacob Lurie, Higher Topos Theory

The enriched situation is section A.3.3 there.