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November 23, 2006

The 1-Dimensional 3-Vector Space

Posted by Urs Schreiber

I feel a certain need for 3-vector spaces, for 3-reps of 3-groups on 3-vector spaces. And things like that. But 1-dimensional 3-vector spaces would do.

Here I shall talk about how, for any braided abelian monoidal category CC, the 3-category

(1)Alg(C):=Σ(Bim(C)) Alg(C) := \Sigma(\mathrm{Bim}(C))

plays the role of the 3-category of canonical 1-dimensional 3-vector spaces.

Moreover, I would like to point out how morphisms between almost-trivial line-3-bundles with connection give rise to the 3-category of twisted bimodules that I talked about recently #.

This 3-category is a beautiful gadget. For C=Mod RC = \mathrm{Mod}_R and RR any commutative ring,

(2)Alg(Mod R) \mathrm{Alg}(\mathrm{Mod}_R)

is discussed in the last part of

R. Gordon, A.J. Power and R. Street,
Coherence for tricategories,
Memoirs of the American Math. Society 117 (1995) Number 558.

John Baez describes this guy in TWF 209.

I first got interested in it here, but for a dumb reason it took me until last night to realize that this is the 3-category of canonical 1-dimensional 3-vector spaces that I was looking for all along.

For reading on, you have to leave the room and go to this file:

\;\;\;\;the 1-dimensional 3-vector space

Posted at November 23, 2006 3:14 PM UTC

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