Quillen bifunctor



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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.



(Quillen bifunctor)

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

  1. for any cofibration i:cci : c \to c' in CC and cofibration j:ddj : d \to d' in DD, the induced (pushout product) morphism

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

    is a cofibration in EE, which is a weak equivalence if either ii or jj is a weak equivalence

  2. it preserves colimits separately in each variable


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

F(c,d) F(Id,j) F(c,d) F(i,Id) F(c,d) F(c,d) F(c,d)F(c,d). \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=(c)i = (\emptyset \hookrightarrow c) we have F(,d)=F(,d)=F(\emptyset, d) = F(\emptyset, d') = \emptyset (since the initial object is the colimit over the empty diagram and FF is assumed to preserve colimits) and the above pushout diagram reduces to

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

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



Let :C×DE\otimes : C \times D \to E be an adjunction of two variables between model categories and assume that CC and DD 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 1c 2f : c_1 \to c_2 and g:d 1d 2g : d_1 \to d_2 we have for the morphism

(c 1d 2) c 1d 1(c 2d 1)c 2d 2 (c_1 \otimes d_2) \coprod_{c_1 \otimes d_1} (c_2 \otimes d_1) \to c_2 \otimes d_2


  • a cofibration if both ff and gg 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


Monoidal and enriched model categories

Lift to coends over tensors

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


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

Then the coend functor

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

is again a ´Quillen bifunctor.

This Lurie, prop. A.2.9.26 with remark A.2.9.27.

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

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

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

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

One possible choice is

Q proj(*)=N(/C) op. 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=Δ opC = \Delta^{op} this is the classical choice by Bousfield and Kan, see Bousfield-Kan map.

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

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

For C=ΔC = \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

Last revised on February 2, 2021 at 23:16:08. See the history of this page for a list of all contributions to it.