Finn Lawler empty 4 (Rev #2, changes)

Showing changes from revision #1 to #2: Added | Removed | Changed

I want to define a notion of cartesian equipment, the idea being that a bicategory $M$ should be a cartesian bicategory if and only if the proarrow equipment $Map M \to M$ is a cartesian equipment.

• See

  Wood’s: an identity-on-objects, locally fully faithful pseudofunctor $K \to M$, written $f \mapsto f_*$, where each $f_*$ has a right adjoint $f^*$ in $M$.

  equipment? first.

• Shulman’s: a framed bicategory, that is, a pseudo double category $M$ whose source and target functors $(s,t) \colon M_1 \to M_0 \times M_0$ form a bifibration.

• that of Carboni–Kelly–Verity–Wood: a (normal) pseudofunctor $M \colon K^{op} \times K \to Cat$, where $K$ is a 1-category.

The With last is strictly more general than the others, correct as notion of equipment, there are even two their possible notions of adjoint morphisms, and hence limits: those ofstarred pointed equipmentsCarboni–Kelly–Verity–Wood , i.e. and those equipped of with a transformation$\hom_K \Rightarrow M$Grandis–Paré , where (for the double left categories). and right actions of$K$ have suitable adjoints. These CKVW equipments should be equivalent to the others if we ask for a transformation $M^2 \Rightarrow M$ making $M$ a (pseudo)monad in a suitable bi- or tricategory of ‘biprofunctors’.

Question: Do these notions coincide?

Then a cartesian equipment will be an equipment that ‘has finite products’, that is an equipment $M$ for which the diagonals $M \to M^2$ and $M \to 1$ have right pseudoadjoints.