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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?

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1 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.

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