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There are lots of things to say, but here's one: In article <412b269b@news.sentex.net>, Urs Schreiber wrote: >So I am >wondering if by reverse engineering this construction from loop space >language to algebraic higher-group theory, one can find a previously >unnoticed sort of "2-group" or something similar which does not have $>B = -F$ but has "quasi-abelian" B and non-abelian A. A strict 2-group is the same as a crossed module: a group G, a group H, an action of G on H and a homomorphism t: $H -> G,$ satisfying some equations described in my paper. Taking Lie algebras we see a strict Lie 2-algebra is the same as a differential crossed module: a Lie algebra g, a Lie algebra h, an action of g on h and a homomorphism dt: $h -> g,$ satisfying some equations. When we think about connections in this context, we see the natural equation is not $$B = -F$$ (which makes no sense!) but instead t(B) $= -F$ where F is a g-valued 2-form and B is an h-valued 2-form. We can then consider various special cases. At one extreme we can have G trivial and H abelian; then the above equation is vacuous. This is what happens in 2-form electromagnetism. At another extreme we can have $G = H$ and t the identity; then you get $B = -F$. That's the case you seem to like best. But there are lots of intermediate cases. Maybe you want some concrete examples? I can manufacture examples, but not very interesting ones if G and H are required to be *compact* Lie groups, because the Lie algebra of these is always semisimple + abelian, and the options for homomorphisms dt are severely limited. (As usual I will cc this to you.)