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Do the twist! Do the shift! Do the Poincare 2-group shuffle!
I understand your puzzlement. But maybe you should have said "why not just the translation group"? The objects are Lorentz transformations; to specify a morphism we also need a translation.
The point is this: the set of morphisms from any object to itself is just the translation group, but the set of all morphisms is the Poincare group!
Consider the set of all morphisms. Any morphism looks like
this:
t --ts--> t
To specify it, we need to give the pair (t,s), which is an element of the Poincare group.
Next fix an object t, and consider the set of morphisms from t to itself. Any such morphism looks like this:
t --ts--> t
Just like before! But now we've fixed t ahead of time, so the only thing we get to choose is s, which is an element of the translation group.
In the crossed module approach to 2-groups, we get the Poincare 2-group by taking G = Lorentz group and H = translation group, with the obvious action of G on H. Then the group of all morphisms is the semidirect product of G on H, which is the Poincare group.
In other words, as someone said a while back, we're thinking about the Poincare group as built out of Lorentz transformations (twists) and translations (shifts) in the usual way. But now, we're thinking of the twists as objects and the shifts as automorphisms (morphisms from an object to itself).
selfAdjoint said:So (once more into the breach...) in the Lorentz/Poincare example, the objects that you can multiply are the rotations and boosts ("twists") for the Lorentz group, and the morphisms you can multiply come from the elements of the Poincare group ("twist-shifts"), and you said that a twist-shift ts defines a morphism from object t to object t.
So now my question is, why do we say the morphisms are described by the Poincare group; why not just the Lorentz group?
I understand your puzzlement. But maybe you should have said "why not just the translation group"? The objects are Lorentz transformations; to specify a morphism we also need a translation.
The point is this: the set of morphisms from any object to itself is just the translation group, but the set of all morphisms is the Poincare group!
Consider the set of all morphisms. Any morphism looks like
this:
t --ts--> t
To specify it, we need to give the pair (t,s), which is an element of the Poincare group.
Next fix an object t, and consider the set of morphisms from t to itself. Any such morphism looks like this:
t --ts--> t
Just like before! But now we've fixed t ahead of time, so the only thing we get to choose is s, which is an element of the translation group.
In the crossed module approach to 2-groups, we get the Poincare 2-group by taking G = Lorentz group and H = translation group, with the obvious action of G on H. Then the group of all morphisms is the semidirect product of G on H, which is the Poincare group.
In other words, as someone said a while back, we're thinking about the Poincare group as built out of Lorentz transformations (twists) and translations (shifts) in the usual way. But now, we're thinking of the twists as objects and the shifts as automorphisms (morphisms from an object to itself).