Re: biadjoint biequivalences and spans in 2-categories
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
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
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
participants (3)
-
Michael Shulman -
Richard Garner -
Tom Fiore