Re: biadjoint biequivalences and spans in 2-categories

[Tom Fiore <fiore@math.uchicago.edu>]

Hello,
 	Details of a part of John and Mike's post can be found in one of
my publications.

Theorem. 9.17
Let X and A be strict 2-categories, and G:A -> X a pseudo functor. There
exists a left biadjoint for G if and only if for every object x of X there
exists an object r of A and a biuniversal arrow x -> Gr from x to G.

The proof is as one could expect.

http://arxiv.org/abs/math.CT/0408298

Fiore, Thomas M. Pseudo limits, biadjoints, and pseudo algebras:
categorical foundations of conformal field theory. Mem. Amer. Math. Soc.
182 (2006), no. 860, x+171 pp.

Best greetings,
Tom

On Tue, 19 Aug 2008, Michael Shulman wrote:

> Hi John,
>
> Answers to both this and your previous question about biadjoint
> biequivalences are at least asserted in Street's "Fibrations in
> Bicategories".
>
> At the end of section 1, he defines a functor (=homomorphism) to have a
> left biadjoint if each object has a left bilifting, to be a biequivalence
> if it is biessentially surjective and locally fully faithful, and states
> that "clearly a biequivalence T has a left biadjoint S which is also a
> biequivalence".
>
> At the beginning of section 3 he defines a bicategory of spans from A to
> B in any bicategory, and given finite bilimits, essentially describes
> how to construct what one might call an "unbiased tricategory" of spans
> (of course, the definition of tricategory didn't exist at the time).
>
> He doesn't give any details of the proofs, but one could probably
> construct a detailed proof from these ideas without much more than
> tedium.  I don't know whether anyone has written them out.
>
> Best,
> Mike
>
> On Tue, Aug 19, 2008 at 07:45:12AM -0700, John Baez wrote:
>> Dear Categorists -
>>
>> Given a category C with pullbacks we can define a bicategory Span(C)
>> where objects are objects of C, morphisms are spans - composed
>> using pullback - and 2-morphisms are maps between spans.
>>
>> Have people tried to categorify this yet?
>>
>> Suppose we have a 2-category C with pseudo-pullbacks.  Then we should
>> be able to define a tricategory Span(C).   Has someone done this?
>>
>> Or maybe people have gotten some partial results, e.g. in the case
>> where C = Cat.  I'd like to know about these!
>>
>> Best,
>> jb
>>
>>
>>
>>
>
>