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Idea

A (left) Quillen bifunctor is a functor of two variables between model categories that respects combined cofibrations in its two arguments in a suitable sense.

The notion of Quillen bifunctor enters the definition of monoidal model category and of enriched model category.

Definition

Definition

(Quillen bifunctor)

Let $C, D, E$ be model categories. A functor $F : C \times D \to E$ is a Quillen bifunctor if it satisfies the following two conditions:

1. for any cofibration $i : c \to c'$ in $C$ and cofibration $j : d \to d'$ in $D$, the induced (pushout product) morphism

$F(c', d) \coprod_{F(c,d)} F(c,d') \to F(c', d')$

is a cofibration in $E$, which is a weak equivalence if either $i$ or $j$ is a weak equivalence

2. it preserves colimits separately in each variable

Remarks

In full detail the pushout appearing in the first condition is the one sitting in the pushout diagram

$\array{ F(c,d) &\stackrel{F(Id,j)}{\to}& F(c,d') \\ \;\;\downarrow^{F(i,Id)} && \downarrow \\ F(c',d) &\stackrel{}{\to}& F(c', d) \coprod_{F(c,d)} F(c,d') } \,.$

In particular, if $i = (\emptyset \hookrightarrow c)$ we have $F(\emptyset, d) = F(\emptyset, d') = \emptyset$ (since the initial object is the colimit over the empty diagram and $F$ is assumed to preserve colimits) and the above pushout diagram reduces to

$\array{ \emptyset &{\to}& \emptyset \\ \;\;\downarrow && \downarrow \\ F(c,d) &\stackrel{}{\to}& F(c,d) } \,.$

Therefore for $c$ a cofibrant object the condition is that $F(c,-) : D \to E$ preserves cofibrations and acyclic cofibrations. Similarly for $d$ fibrant the condition is that $F(-,d) : C \to E$ preserves cofibrations and acyclic cofibrations.

Properties

Proposition

Let $\otimes : C \times D \to E$ be an adjunction of two variables between model categories and assume that $C$ and $D$ are cofibrantly generated model categories. Then $\otimes$ is a Quillen bifunctor precisely if it satisfies its axioms on generating (acyclic) cofibrations, i.e. if for $f : c_1 \to c_2$ and $g : d_1 \to d_2$ we have for the morphism

$(c_1 \otimes d_2) \coprod_{c_1 \otimes d_1} (c_2 \otimes d_1) \to c_2 \otimes d_2$

is

• a cofibration if both $f$ and $g$ are generating cofibrations;

• an acyclic cofibration if one is a generating cofibration and the other a generating acyclic cofibration.

This appears for instance as Corollary 4.2.5 in

Applications

Monoidal and enriched model categories

• In a monoidal model category $C$ the tensor product $\otimes : C \times C \to C$ is required to be a Quillen bifunctor.

• An enriched model category $D$ over the monoidal model category $C$ is one that is powered and copowered over $D$ such that the copower $\otimes : D \times C \to D$ is a Quillen bifunctor.

Lift to coends over tensors

The following proposition asserts that under mild conditions a Quillen bifunctor on $C \times D$ lifts to a Quillen bifunctor on functor categories of functors to $C$ and $D$.

Proposition

Let $\otimes : C \times D \to E$ be a Quillen functor. Let

• $S$ be a Reedy category and take the functor categories $[S,C]$ and $[S^{op},C]$ be equipped with the correspondingReedy model structure.

• or assume that $C$ and $D$ are combinatorial model categories and let $[S,C]$ and $[S^{op},A]$ be equipped, respectively with the projective and the injective globel model structure on functor categories.

Then the coend functor

$\int^{S} (- \otimes -) : [S,C]\times [S^{op},D] \to E$

is again a ´Quillen bifunctor.

It follows that the corresponding left derived functor computes the corresponding homotopy coend.

Bousfield-Kan type homotopy colimits

This is an application of the above application.

Let $C$ be a category and $A$ be a simplicial model category. Let $F : C \to A$ be a functor and let ${*} : C^{op} \to A$ be the functor constant on the terminal object.

Consider the global model structure on functors $[C^{op},SSet]_{proj}$ and $[C^{op},A]_{inj}$ and let $Q({*})_{proj}$ be a cofibrant replacement for ${*}$ in $[C^{op},Set]_{proj}$ and $Q_{inj}(F)$ a cofibrant replacement for $F$ in $[C,A]_{inj}$.

One show that the homotopy colimit over $F$ is computed as the coend or weighted limit

$hocolim F = \int Q_{proj}({*}) \cdot Q_{inj}(F) \,.$

One possible choice is

$Q_{proj}({*}) = N(-/C)^{op} \,.$

That this is indeed a projectively cofibrant resulution of the constant on the terminal object is for instance proposition 14.8.9 of

• Hirschhorn, Model categories and their localization .

For the case that $C = \Delta^{op}$ this is the classical choice by Bousfield and Kan, see Bousfield-Kan map.

Assume that $A$ takes values in cofibrant objects of $A$, then it is already cofibrant in the injective model structure $[C,A]_{inj}$ on functors and we can take $Q_{inj}(F) = F$. Then the above says that

$hocolim F = \int N(-/C)^\op \cdot F \,.$

For $C = \Delta$ this is the classical prescription by Bousfield-Kan for homotopy colimits, see also the discussion at weighted limit.

Using the above proposition, it follows in particular explicitly that the homotopy colimit preserves degreewise cofibrations of functors over which it is taken.

A nice discussion of this is in (Gambino).

Appendix A.2 of