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In Bénabou's 1967 Introduction to Bicategories, he mentions that, in a forthcoming part II, he would study bicategories $\mathcal K$ in which each composition functor $$\circ_{A, B, C} \colon \mathcal K(A, B) \times \mathcal K(B, C) \to \mathcal K(A, C)$$ admits a right adjoint. Sadly, a part II never appeared. Has this notion been studied anywhere elsewhere in the literature?

Two remarks:

  • When $\mathcal K$ has a single object (hence is a monoidal category), this condition is equivalent to asking for the monoidal category to be cocartesian (see here). Consequently, bicategories satisfying this condition are one possible horizontal categorification of categories with finite coproducts.
  • Bicategories satisfying the weaker property of every composition functor admitting a right multiadjoint have been studied by Walker in Generic bicategories (2018). This is the closest to a reference I have found.
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  • $\begingroup$ Nice question. Do you know of any examples? Beyond the monoidal case, I can’t think of any at all, off the top of my head. E.g. even for typical bicategories of sup-complete posets, where composition preserves sups in each variable individually and hence has right adjoints on each side, it doesn’t preserve them as a functor of two variables together, if I’m not mistaken, so can’t have a total right adjoint. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Dec 4 at 10:31
  • $\begingroup$ I must admit I do not know of any examples beyond the one-object case! The condition seems rather strong, which suggests to me that there probably aren't many natural examples, but I believe there must be some (especially if Bénabou thought them interesting enough to study). I would hope that, if a reference exists, it also contains examples, but I would also be interested in an example even without a reference. $\endgroup$
    – varkor
    Commented Dec 4 at 10:44
  • $\begingroup$ Reminds me of the interpolation property for continuous lattices (or posets or categories). $\endgroup$
    – Paul Taylor
    Commented Dec 5 at 13:14

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