Question

When considering functors between model categories one possibility is to restrict ones attention to quillen adjunctions. But what about distributors?

What are the natural distributors to consider between model categories?

Answer 1

There's a natural generalization of Quillen adjunction.

Let $A$ and $B$ be model categories, and $M:A^{op}\times B\to Set$ be a distributor/profunctor/module. This should be Quillen if whenever $i:a\to a'$ is a cofibration in $A$ and $p:b\to b'$ is a fibration in $B$, with either $i$ or $p$ a weak equivalence, then the induced map from $M(a',b)$ to the pullback $M(a,b)\times_{M(a,b')}M(a',b')$ is surjective.