Finn Lawler empty 4 (Rev #2, changes)

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

This shouldn’t be hard to do. There are three notions of equipment in the literature:

Outline

  • See

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

    equipment? first.
  • Shulman’s: a framed bicategory, that is, a pseudo double category MM whose source and target functors (s,t):M 1M 0×M 0(s,t) \colon M_1 \to M_0 \times M_0 form a bifibration.

  • that of Carboni–Kelly–Verity–Wood: a (normal) pseudofunctor M:K op×KCatM \colon K^{op} \times K \to Cat, where KK 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 transformationhom KM\hom_K \Rightarrow MGrandis–Paré , where (for the double left categories). and right actions ofKK have suitable adjoints. These CKVW equipments should be equivalent to the others if we ask for a transformation M 2MM^2 \Rightarrow M making MM a (pseudo)monad in a suitable bi- or tricategory of ‘biprofunctors’.

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 CatProfCat{-}Prof. The Grothendieck construction for such a profunctor should give a double category whose underlying span is a two-sided fibration.

With the correct notion of equipment, there are two possible notions of adjoint morphisms, and hence limits: those of CKVW, and those of Grandis–Paré (for double categories).

Question: Do these notions coincide?

Then a cartesian equipment will be an equipment that ‘has finite products’, that is an equipment MM for which the diagonals MM 2M \to M^2 and M1M \to 1 have right pseudoadjoints.

Revision on May 13, 2011 at 12:14:02 by Finn Lawler?. See the history of this page for a list of all contributions to it.