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• CommentRowNumber1.
• CommentAuthormaxsnew
• CommentTimeMar 15th 2017

I just worked through the definition of double profunctor and was really surprised that when you view a double profunctor as a lax functor $P : C^{\text{op}} \times D \to \text{Span}$, that $\text{Span}$ really means the transpose of what you’d expect if you think of vertical arrows as being function-like and horizontal arrows as being relation-like, and that $\text{op}$ meant horizontal reversal.

I think I’ll change the page so that $\text{Span}$ has functions as vertical arrows and write the lax functor as $P : (C^{\text{co}}\times D)^{T} \to \text{Span}$, so that $\text{co}$ means horizontal reversal. The presence of the transpose then looks very presheaf-like, and I can’t tell if that’s a misleading intuition or not. Also, I think this would obviate the need to define a “vertically lax” functor as mentioned later in the article.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeMar 15th 2017

I don’t agree. Not everyone thinks of vertical arrows as being function-like and horizontal arrows as being relation-like.

• CommentRowNumber3.
• CommentAuthormaxsnew
• CommentTimeMar 15th 2017

That might be true, but I think it would be better for readers if we stuck to a particular convention, for instance the very closely related double category $\text{Prof}$ is defined in double category as having profunctors as horizontal arrows.

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeMar 16th 2017

I suppose there’s some sense in that, although the nLab is not overall consistent on many things. (In an ideal world, I think I would prefer to use informative words like “tight” and “loose” for the two directions of a pseudo double category; then everyone can draw their diagrams however they like but still use the same words for the same things. However, we probably shouldn’t try to change the world so radically yet.)

However, it would be even more consistent to swap vertical and horizontal throughout the entire page, so that it defines a “vertical double profunctor” rather than a “horizontal one”. For instance, lower down on the current page we have to consider “vertically virtual” double categories, whereas the page virtual double category treats the virtual direction as horizontal.

• CommentRowNumber5.
• CommentAuthormaxsnew
• CommentTimeMar 16th 2017

Ok, well I don’t really understand the virtual stuff, so I won’t make change those for fear of making it more inconsistent.

In other news I added Hom as an example and added some to the idea section using the language of heteromorphisms.