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July 3, 2008

A Small Observation

Posted by David Corfield

Urs defined the Schreiber 2-group of linear automorphisms of a skeletal Baez-Crans 2-vector space back here. For the space δ=0:F nF m\delta = 0: F^n \to F^m, it has GL(n,F)×GL(m,F)GL(n, F) \times GL(m, F) as objects and F nmF^{n \cdot m} worth of arrows from an object to itself. Arrows are linear maps F nF mF^n \to F^m, and the group of objects acts on them by a kind of conjugation.

Now, the small thought occurred to me that interchanging the mm and nn makes little difference. So when I suggested that the Poincaré 2-group was a sub-2-group of the 2-group for n=1,m=4n = 1, m = 4, I might also have said n=4,m=1n = 4, m = 1.

But all this is not so surprising, as this area is quite span-ish and bimodule-esque.

Posted at July 3, 2008 4:35 PM UTC

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Re: A Small Observation

If I understand the question correctly, I had met the object some years previously. At the same time that John and Alissa were writing their stuff, my student Magnus was finishing his thesis: Representations of crossed modules and cat1-groups, available from
http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/04/algtop04.html#04.05

He did quite a few calculations of the automorphisms of short chain complexes of vector spaces (i.e. BC2VSpaces, which perhaps should be BCFB2VSpaces!) as these were the crossed modules that, in his theory, took the place of the vector spaces of ordinary representation theory. There is a lot of nice stuff in that thesis resulting from long hours of work by Magnus (and, of course, his supervisor). Looking back at some of the discussion in the blog, it may be worth glancing back at Magnus’ work. It was mentioned by John in TWF, at the time, but may have slipped from peoples view since. As it is freely available …..

Posted by: Tim Porter on July 3, 2008 7:36 PM | Permalink | Reply to this

Re: A Small Observation

OK, so it’s no longer the Schreiber 2-group, but rather the Forrester-Barker 2-group. (I dare say Urs will go on to give his name to other things!)

In fact, the symmetries of 2-term complexes are treated in full generality in Chapter 2. For the case of the zero map see section 2.2.3.

Posted by: David Corfield on July 3, 2008 8:16 PM | Permalink | Reply to this

Re: A Small Observation

I would suggest `the self equivalence 2-group’ in honour of that great mathematician Augustus Self-Equivalence!

Posted by: Tim Porter on July 3, 2008 8:53 PM | Permalink | Reply to this

Re: A Small Observation

Wait, there is a long and honored tradition to never ever name a concept after the person who actually first discovered it. You should stick to that!! ;-)

Posted by: Urs Schreiber on July 4, 2008 10:15 AM | Permalink | Reply to this

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