The locally model bicategory of $\cal V$-profunctors

Question

Not related to this old question of mine, but takes the question from a different perspective.

Let $\mathcal V$ be a monoidal model category (following the def of Hovey, for example). Then there is a bicategory $\text{Prof}(\mathcal V)$ of $\cal V$-valued profunctors, which has the following interesting property:

every hom-category has a model structure[1].

This is, I think, the paradigmatic example of a "locally model bicategory". There can be others: I'm interested in examples for this notion and related results.

I'm trying to get a list of properties to impose to the general notion (a "locally model 2-category", i.e. a 2-category "enriched" over model categories -this is not a true definition, as there is no sensible monoidal structure on model categories-).

Let's for example consider the following explicit question:

Let $\varphi\colon {\bf A}\looparrowright{\bf B}$ be a profunctor, and $\bf X$ be a category; then precomposition by $\varphi$ gives $$ \text{Prof}({\bf B},{\bf X}) \overset{-\diamond \varphi}\to \text{Prof}({\bf A},{\bf X}) $$ which has left and right adjoints $\text{Lan}_\varphi$, $\text{Ran}_\varphi$ defined by the co/ends $$ \text{Ran}_\varphi\psi(b,x) \cong \int_a \hom(\varphi(a,b), \psi(a,x)) $$ (Lan is similar). Does $-\diamond \varphi \dashv \text{Ran}_\varphi$ form a Quillen adjunction?

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[1] in fact, many! Let's take $\mathcal V = \bf sSet$ and declare that I want to study the injective model structure on $\text{Prof}(\mathbf{sSet})(\mathbf A,\mathbf B)=[\mathbf A^\text{op}\times \mathbf B,\mathbf{sSet}]$.

Answer 1

Given two model categories $\mathcal{M},\mathcal{N}$, one does know what would have been a left Quillen functor out of what would have been the tensor product $\mathcal{M} \otimes \mathcal{N}$ into a third model category $\mathcal{K}$, and that is a left Quillen bifunctor $\mathcal{M} \times \mathcal{N} \to \mathcal{K}.$ This leads to a natural notion of a category (weakly) enriched in model categories, namely, a bicategory $\mathcal{C}$ such that each mapping category $\mathrm{Map}_{\mathcal{C}}(X,Y)$ carries a model structure and such that each composition operation $$ \mathrm{Map}_{\mathcal{C}}(X,Y) \times \mathrm{Map}_{\mathcal{C}}(Y,Z) \to \mathrm{Map}_{\mathcal{C}}(X,Z) $$ is a left Quillen bifunctor. In the case of profunctors this can be achieved, for example, if one endows $\mathrm{Fun}(\mathbf{A}^{\mathrm{op}} \times \mathbf{B},\mathcal{V})$ with the model structure which is injective in the $\mathbf{A}^{\mathrm{op}}$ coordinate and projective in the $\mathbf{B}$ coordinate (call it the injective-projective model structure). Note that if one defines enrichment in model categories in this way then pre-composition and post-composition with a cofibrant morphism is a left Quillen functor. In particular, pre-composition or post-composition with a cofibrant profunctor $\varphi:\mathbf{A} \looparrowright \mathbf{B}$ (with respect to the injective-projective model structure) induces a left Quillen functor $\text{Prof}({\bf B},{\bf X}) \overset{-\diamond \varphi}\to \text{Prof}({\bf A},{\bf X})$ (both equipped with the injective-projective model structure).