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Jun23-06, 01:27 AM
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Quote Quote by john baez
It's hot up here in Waterloo, Canada!

In the Poincare 2-group, the group of objects is the group of Lorentz
transformations, and the group of morphisms is the Poincare group.

That doesn't completely describe the Poincare 2-group. You need to
know some other stuff, like:

If you have a morphism in here, which object does it start at, and
which object does it end at?

There's only one sensible answer to this question, I think, so I'll leave it as a puzzle.

You also need to decide how to compose morphisms. I'll leave that as a (harder) puzzle.
About the puzzle---I think selfAdjoint has already figured it out but I did not follow all the discussion, so I will take a guess.

Someone, I think it was JB, suggested using letters T and S for "twist" and "shift", where T is an element of the Lorentz group and S is thought of as a translation (we are building the Poincaré group)

[EDIT] GROAN. I just looked at the "Higher Yang Mills" paper that I lost a couple of days ago, while cleaning up the living room. the Poincaré 2-group is explained as EXAMPLE 9. So the puzzle was already answered.

...I talked to Kea and she suggested one by Girelli and Pfeiffer, which I never got around to looking at it. Maybe I will now.
Higher gauge theory -- differential versus integral formulation
Florian Girelli, Hendryk Pfeiffer
26 pages
J.Math.Phys. 45 (2004) 3949-3971
"The term higher gauge theory refers to the generalization of gauge theory to a theory of connections at two levels, essentially given by 1- and 2-forms. So far, there have been two approaches to this subject. The differential picture uses non-Abelian 1- and 2-forms in order to generalize the connection 1-form of a conventional gauge theory to the next level. The integral picture makes use of curves and surfaces labeled with elements of non-Abelian groups and generalizes the formulation of gauge theory in terms of parallel transports..."

I hope Kea is better now and was wishing she would suddenly materialize amidst this thread.

Well, looking at Girelli/Pfeiffer, I see right away references
[15] J. C. Baez: Higher Yang–Mills theory (2002). Preprint hep-th/0206130.
[19] J. C. Baez and A. Crans: Higher dimensional algebra VI: Lie 2-algebras (2003). Preprint math.QA/0307263.

My posts are just not helpful in this thread at this point. I shall delete the next one.