I seem to recall that there is a straightforward subfactor construction that yields fusion categories given by G-graded vector spaces and representations of G, for finite groups G. Is there an analogous construction for 2-groups?


Some background: A 2-group is a monoidal groupoid, for which the isomorphism classes of objects form a group. Sinh showed that up to monoidal equivalence, these are classified by a group G (isom. classes of objects), a G-module H (automorphisms of identity), and an element of H3(G,H). In the context of this discussion, we can limit our attention to G finite, H=Cx. One notable feature is that when the action of G on H is trivial, the three-cocycle twists the associator in the G-graded vector space category.


I'm mostly curious about how to tell when two elements of H3(G,H) yield Morita-equivalent fusion categories, and am wondering if subfactors or planar algebras make it easy to detect this.



Is there a subfactor construction involving 2-groups?
This is a standard construction in Subfactor theory see the intro of <http://arxiv.org/abs/0811.1084v2> for details. The construction goes back a long long way (if I remember correctly both Vaughan Jones and Adrian Ocneanu's theses were related to this question, but I could be wrong there).


From a category theory perspective recall that a subfactor (N < M) is a unitary tensor category C (the N-N bimodules) together with a Frobenius algebra object A in C (M as an N-N bimodule with conditional expectation as trace). In this case the tensor category C is the twisted category of G-graded vector spaces (where you use the 3-cocycle to change the associator), and the algebra object is a twisted version of the group algebra (or maybe just the group algebra? I'm getting confused, shouldn't group algebras be twisted by 2-cocycles?).


