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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeApr 2nd 2010

Wrote a bit at double profunctor.

• CommentRowNumber2.
• CommentAuthorunmuamua
• CommentTimeJul 25th 2018
• (edited Jul 25th 2018)
Hello! ncatlab "discussion" link sends me here.

I do not see any references to papers at the double profunctor page and I am not able to google any. The question of interest is about existence of the composite (i.e. the proposition of (non)-representability of virtual equipment DblProf), mentioned there as "Composition of double profunctors is, unfortunately, hard to define and not well-behaved." Are there any explicit counterexamples of non-existence or "not well-behaved"?

I am not sure whether I should ask the question an mathoverflow, but it would be great to see some info at the ncatlab page.
1. Hi unmuamua, just to say that it’s great that you posted your question here, and that you followed the discussion link to here; that is exactly the intended purpose! I’m sure Mike or someone else knowledgeable will take a look at your question soon.

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeJul 25th 2018

Hi! There are no other references that I know of; that page is original research. (To link to an nLab page in an nForum post, put its name in double square brackets, like [[double profunctor]].) The two subsections under “Composition” describe two ways to define composition, so composites of a sort certainly do “exist”. The ill-behavedness referred to is more explicitly that neither of the two composites mentioned satisfies the necessary conditions to be associative. I haven’t worked out a specific example of double profunctors $H,K,L$ such that $(H\circ K)\circ L \neq H\circ (K\circ L)$, but you could probably do it as an exercise by starting with the relevant input counterexample (either of a coequalizer in $Cat$ not preserved by pullback or a bo-ff factorization in $DblCat$ not preserved by pushout). If you do, please add it to the nLab page!

• CommentRowNumber5.
• CommentAuthorunmuamua
• CommentTimeJul 25th 2018
Thank you!
• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeOct 19th 2022

Added a concrete example of the failure of associativity for composition of double profunctors.

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