Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
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.
Let be model categories. A functor is a Quillen bifunctor if it satisfies the following two conditions:
for any cofibration in and cofibration in , the induced (pushout product) morphism
is a cofibration in , which is a weak equivalence if either or is a weak equivalence
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
In particular, if we have (since the initial object is the colimit over the empty diagram and is assumed to preserve colimits) and the above pushout diagram reduces to
Therefore for a cofibrant object the condition is that preserves cofibrations and acyclic cofibrations. Similarly for cofibrant the condition is that preserves cofibrations and acyclic cofibrations.
Let be an adjunction of two variables between model categories and assume that and are cofibrantly generated model categories. Then is a Quillen bifunctor precisely if it satisfies its axioms on generating (acyclic) cofibrations, i.e. if for and we have for the morphism
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 lifts to a Quillen bifunctor on functor categories of functors to and .
Let be a Quillen functor. Let
Then the coend functor
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 be a category and be a simplicial model category. Let be a functor and let be the functor constant on the terminal object.
Consider the global model structure on functors and and let be a cofibrant replacement for in and a cofibrant replacement for in .
One show that the homotopy colimit over is computed as the coend or weighted limit
One possible choice is
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 this is the classical choice by Bousfield and Kan, see Bousfield-Kan map.
Assume that takes values in cofibrant objects of , then it is already cofibrant in the injective model structure on functors and we can take . Then the above says that
For 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