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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 be model categories. A functor (out of the product category of with ) is a Quillen bifunctor if it satisfies the following two conditions:
for any
cofibration in
cofibration in ,
the induced pushout product-morphism
is a cofibration in , which is a weak equivalence if either or is a weak equivalence.
In more detail, the pushout appearing in the first condition in Def. is the one sitting in the following pushout square:
In particular, if (for denoting the initial object) 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;
for a cofibrant object, 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
is
a cofibration if both and 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
In a monoidal model category the tensor product is required to be a Quillen bifunctor.
An enriched model category over the monoidal model category is one that is powered and copowered over such that the copower is a Quillen bifunctor.
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
be a Reedy category and take the functor categories and be equipped with the correspondingReedy model structure.
or assume that and are combinatorial model categories and let and be equipped, respectively with the projective and the injective globel model structure on functor categories.
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.
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 shown in Hirschhorn (2002), Prop 14.8.9.
For the case that (the opposite of the simplex category) this is the classical choice in the discussion of the 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 (2010).
Mark Hovey, Def. 4.2.1 in: Model Categories, Mathematical Surveys and Monographs, 63 AMS (1999) [ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books]
Jacob Lurie, Appendix A.2 of: Higher Topos Theory (2009)
Nicola Gambino, Def. 2.3 in: Weighted limits in simplicial homotopy theory, Journal of Pure and Applied Algebra 214 7 (2010) 1193–1199 [doi:10.1016/j.jpaa.2009.10.006]
Last revised on May 21, 2023 at 12:39:08. See the history of this page for a list of all contributions to it.