Is the enriched bicategory of $V$-profunctors considered in the literature anywhere? Here I want $V$ to be, say, a complete, cocomplete, closed, symmetric monoidal category; and by enriched bicategory I mean $V$-enriched at the 2-cell level, or, equivalenty, $V$-cat enriched at the hom level.

At the enriched bicategory page, three of the interesting looking references are seemingly unavailable, namely Sean Carmody’s thesis, Steve Lack’s thesis and Alex Hoffnung’s notes. Does anyone know if these are online anywhere? (I realise I could email Steve or Alex, but having links from the nlab page would be even better.)

This ought to be a protoytpical example of a $V$-cat enriched bicategory, and it would be nice if someone has already proved it!

]]>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.

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