on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
on strict ∞-categories?
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 $C, D, E$ be model categories. A functor $F : C \times D \to E$ is a Quillen bifunctor if it satisfies the following two conditions:
for any cofibration $i : c \to c'$ in $C$ and cofibration $j : d \to d'$ in $D$, the induced (pushout product) morphism
is a cofibration in $E$, which is a weak equivalence if either $i$ or $j$ 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 $i = (\emptyset \hookrightarrow c)$ 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
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.
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
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
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.
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 $D$.
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.
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 $C$ 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
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
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
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
Nicola Gambino, Weighted limits in simplicial homotopy theory (pdf)