Model bicategories

Question

From a conceptual point of view, the notion of a "model bicategory" is clear: a complete, cocomplete bicategory where there are two "very weak" factorization systems, where the commutativity of the squares you all know holds only up to specified invertible 2-cells, as in

I expect this notion to be either pervasive like the classical one, or empty.

What is the case? Is it possible to retrieve "familiar bimodel structures" in places which are bicategorical either because we are weakening a strict structure (categories + lax functors) or rather because their composition law is intrinsically defined up to a controllable isomorphism (spans, profunctors, presheaves with the convolution product induced by a promonoidal structure...)?

If this hasn't been done (like a rapid googling seems to suggest), where is the problem? Even forgetting any topological motivations, are you able to explain me why I shouldn't care about such a weakening?

Answer 1

Whether or not there are nontrivial examples, I would expect such a notion to be less useful than the classical one, because one of the points of the classical definition is to be able to use strict 1-categorical limit/colimit constructions to present homotopical information, whereas in a bicategory you're already being forced to use notions that are up-to-homotopy one dimension higher. Moreover, since bicategories can be strictifed into 2-categories, I suspect that in most cases where there might be a model bicategory, there is also a model 2-category which falls under the classical theory, and that anyone interested in a strict model for the objects in question is using the latter instead.

Answer 2

Funny that someone upped this question of mine today. After ten years I can say that yes: there are at least three examples of "model bicategory", each irreducible to one another, but in a sense different than the one I naively thought in 2014. The question stayed in the back of my head since, and floated back up recently (at ct24 in Santiago).

A model bicategory is a bicategory $\cal K$ such that

• each hom-category ${\cal K}(A,B)$ is a model category;
• each composition functor ${\cal K}(B,C)\times {\cal K}(A,B) \to {\cal K}(A,C)$ is left Quillen.

Three examples of this definition:

• The bicategory of profunctors enriched over a monoidal model category $\cal V$, iirc with respect to the injective model structure on presheaf-hom-categories;
• the strict 2-category of combinatorial model categories (an example Boris Chorny suggested to me in 2016),
• the category of Mealy or Moore automata valued in a monoidal model category $\cal M$ (see here), using a Crans-style transfer theorem of model structure along the pullback in that nLab page.

I look forward having the time to work on this, but by all means tell me if someone did something similar since 2014!