Finn Lawler empty 5 (Rev #2, changes)

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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.

Question: Why are these all equivalent?

An identity-on-objects (pseudo)functor that is locally fully faithful is essentially the same thing as an identity-on-objects functor out of a locally discrete bicategory. In the case of strict 2-categories, these are (by some enriched-category nonsense) precisely the Kleisli objects for monads (on locally discrete strict 2-categories) in $Cat{-}Prof$ . The Grothendieck construction for such a profunctor should give a double category whose underlying span is a two-sided fibration.

An identity-on-objects (pseudo)functor that is locally fully faithful is essentially the same thing as an identity-on-objects functor out of a locally discrete bicategory. In the case of strict 2-categories, these are (by some enriched-category nonsense) precisely the Kleisli objects for monads (on locally discrete strict 2-categories) in $CatProf$ . So we need to know what a biprofunctor is, and to check that biprofunctors form at least a bicategory (if not an equipment) that has well-behaved Kleisli objects.

The Grothendieck construction for a monad on a 1-category in $BiProf$ should then give a double category whose underlying span is a two-sided fibration.

Revision on May 13, 2011 at 13:14:05 by Finn Lawler?. See the history of this page for a list of all contributions to it.