Finn Lawler empty 5 (Rev #3, changes)

Showing changes from revision #2 to #3: Added | Removed | Changed

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?

An Let’s identity-on-objects forget (pseudo)functor about that is locally fully faithful is essentially the same ‘op-connections’ 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 the (on moment. 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 A Grothendieck pseudofunctor construction for a monad on a 1-category in$\mathrm{<span><del\; class="diffmod">\; BiProf</del><ins\; class="diffmod">\; K</ins></span>}\to M$ BiProf K \to M should that then give a double category whose underlying span is a the two-sided identity fibration. 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 $K$.