Finn Lawler empty 5 (Rev #4, changes)

There are three notions of equipment in the literature:

• 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 $M$.

• 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 last is strictly more general than the others, as are even their starred pointed equipments, i.e. those equipped with a transformation $\hom_K \Rightarrow M$, where the left 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’.

• There is also the related notion of a connection on a double category — an equipment is a (pseudo) double category with both a connection and an ‘op-connection’, the latter giving e.g. the right adjoints $f^*$ in Wood’s definition.

Question: Why are these all equivalent?

Let’s forget about the ‘op-connections’ for the moment.

A pseudofunctor $K \to M$ that is the identity on objects and locally fully faithful is the same as an identity-on-objects pseudofunctor $K' \to M$ out of the locally discrete bicategory given by simply throwing away the 2-cells of $K$.

An identity-on-objects pseudofunctor $K \to M$ exhibits $M$ as the Kleisli object of a monad on $K$ in the 2-category of biprofunctors (q.v.). This will be the associated CKVW equipment.

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