Re: biadjoint biequivalences and spans in 2-categories

Re: biadjoint biequivalences and spans in 2-categories

[Michael Shulman]

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
>
>
>
>

Re: biadjoint biequivalences and spans in 2-categories

[Tom Fiore]

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
>>
>>
>>
>>
>
>

Re: biadjoint biequivalences and spans in 2-categories

[Richard Garner]

> 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.

One amusing thing about the tricategory of spans in a
2-category, with composition by iso-comma object, is that it
is barely a tricategory. Binary composition is associative up
to iso; it is only the unitality which is really up to
equivalence. Something similar happens if you define a
tricategory of biprofunctors.

Richard