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Naive question, why is the equivalence relation necessary in the definition of composition of profunctors: Or if it is not necessary what is the advantage of adding the equivalence relation?

Edited: The remarks on bilinearity, and coherence answered my question. But let's suppose our ctegories are enriched over Vect, I am guess the composition would not be just the tensor product version of the definition, if so why not.

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Just because a Kan extention is a colimit then a quotient of a sum by equivalence relation. Otherwise the composition of simple functors isnt coherent by composition of the associate profunctors (a (simple) functors can view as a profunctor in covariant, or contravariant way). – Buschi Sergio Dec 27 '11 at 8:26
Perhaps one can add that coherence is the norm and strict associativity is almost unnatural. – Tim Porter Dec 27 '11 at 11:11
up vote 3 down vote accepted

Profunctor composition is not strictly associative even after putting on the equivalence relation. One is in a bicatgeory setting here. The equivalence relation is among other things to make identity profunctors work. You can think of profunctors as bimodules and composition as tensor product. The equivalence relation is the usual one for tensor products of bimodules. Associativity upto iso is because tensor product is associative up to isomorphism.

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It may be worth following up Ben's answer by thinking that the equivalence relation is what encodes the 'bilinearity'. If you do not have it you go towards a cartesian product instead of a tensor, ... but that is not strictly associative either. – Tim Porter Dec 27 '11 at 5:57

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