Is there a notion of "good" distributor/profunctor for model categories? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T08:35:27Z http://mathoverflow.net/feeds/question/24722 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/24722/is-there-a-notion-of-good-distributor-profunctor-for-model-categories Is there a notion of "good" distributor/profunctor for model categories? Garlef Wegart 2010-05-15T10:43:07Z 2010-11-18T11:47:37Z <p>When considering functors between model categories one possibility is to restrict ones attention to quillen adjunctions. But what about distributors?</p> <blockquote> <p>What are the natural distributors to consider between model categories?</p> </blockquote> http://mathoverflow.net/questions/24722/is-there-a-notion-of-good-distributor-profunctor-for-model-categories/46474#46474 Answer by Steve Lack for Is there a notion of "good" distributor/profunctor for model categories? Steve Lack 2010-11-18T11:47:37Z 2010-11-18T11:47:37Z <p>There's a natural generalization of Quillen adjunction. </p> <p>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.</p>