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Quillen bifunctor

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Contents

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×DE is a Quillen bifunctor if it satisfies the following two conditions:

  1. for any cofibration i:cc in C and cofibration j:dd in D, the induced 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 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

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) we have F(,d)=F(,d)= (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

F(c,d) F(c,d).\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,):DE preserves cofibrations and acyclic cofibrations. Similarly for d cofibrant the condition is that F(,d):CE preserves cofibrations and acyclic cofibrations.

Applications

Monoidal and enriched model categories

Lift to coends over tensors

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

Proposition

Let :C×DE 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.

Proof

This is proposition A.2.9.26 together with remark A.2.9.27 in

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:CA be a functor and let *:C opA 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

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 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

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

For C=Δ 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

  • Nicola Gambino, Weighted limits in simplicial homotopy theory (pdf)