distributors for 2-categories

[David Roberts <david.roberts@adelaide.edu.au>]

Hi all,

==Background==

I have finally read Jean Benabou's Louvain lectures on distributors, which he
was kind enough to send me earlier this year. In them he describes the following
bicategories of distributors (some paraphrasing/simplification may occur,  and
modernisation of terms - all errors are mine):

Dist
  objects: categories C,D,...
  arrows: functors C^op x D--> Set (equiv. opfibrations H --> C^op x D)
  2-arrows: natural transformations (equiv. cartesian functors over C^op x  D)

Dist(V), for V a symmetric closed monoidal category
  objects: V-enriched categories C,D,...
  arrows: V-functors C^op x D--> V
  2-arrows: V-natural transformations

Dist(E), for E a regular category
  objects: categories internal to E  C,D,...
  arrows: internal opfibrations H --> C^op x D
  2-arrows: internal (cartesian) functors over C^op x D

Dist(K), for K a bicategory
  objects: monads in K . . . .

And here it seems to me the pattern breaks down, as taking as input the
bicategories Cat, V-Cat and Cat(E), we do not arrive at any of the examples on
the previous list.

I understand the motivation behind Dist(K), namely that one considers the
process

E |--> Span(E) |--> Dist(Span(E)) = Dist(E)    (E regular category)

as Cat(E) = Monads(Span(E)). I'm not worried about that too much.

==Question==

1) Has anyone done any work on distributor-like constructions for bicategories
that recover the processes

Cat |--> Dist,
V-Cat |--> Dist(V),
Cat(E) |--> Dist(E)?

Something like universally adding adjoints to all 1-arrows in a bicategory, I
would imagine.

I'm asking this in the context of the equivalence between representable
distributors and anafunctors, so I suppose a secondary question is:

2) Given 1) above, is there a notion of 'representable 1-arrow' in this
universal construction?

Thanks,

David



------------------------------
David Roberts

david.roberts@adelaide.edu.au
University of Adelaide



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