It surprised me that the homotopy category of the canonical model structure on $\text{Cat}$ is its familiar quotient category $\text{Cat}/\sim$ whose morphisms are functors modulo natural transformations.
I can't help but notice that this quotient can be taken in any 2-category $\mathcal C$, where morphisms of $\mathcal C/\sim$ are 1-morphisms modulo 2-morphisms.
- In this general case, is there a model structure on (a complete and cocomplete) $\mathcal C$ whose homotopy category is $\mathcal C/\sim$? Or is this a special property of $\text{Cat}$?
I noticed the following:
- the weak equivalences of $\text{Cat}$ can be mimicked in $\mathcal C$ word-by-word
- isofibrations can be mimicked if there is a notion of "free-standing isomorphism" i.e. an object $i\in\mathcal C$ such that
$
\text{Iso}(c) \cong \mathcal C(i,c)
$
for any $c\in\mathcal C$. - since objects of $\mathcal C$ don't naturally have a notion of objects/morphisms, there is no natural anologue for isocofibrations; we could ask for them to be monomorphisms, but this is a stronger requirement.
Best Answer
In any 2-category $\mathcal K$ whatsoever, you can define an isofibration "representably": it is a morphism $f:x\to y$ such that for every object $a$, the induced functor $\mathcal K(a,x)\to \mathcal K(a,y)$ is an isofibration of categories. Then with your proposal of equivalences as the weak equivalences, there is at most one possible model structure, since the weak equivalences and fibrations determine the cofibrations as those morphisms admitting left lifting against the trivial fibrations.
In fact, as long as $\mathcal K$ is complete and cocomplete as a 2-category, this model structure always exists. The idea for the factorizations is to factor $f:x\to y$ as a cofibration and a trivial fibration using the "mapping cylinder" realized as the pseudo-colimit of $f$, and as a trivial cofibration followed by a fibration using the "path object" given by the pseudo-limit of $f$. This is a theorem of Steve Lack proved in Section 4 here.