Does anyone know where the following useful facts were first proved? A lot of papers just say, "It is known that…" and I’d like to give proper attribution in some future work.
Let A be an abelian group, and let denote the monoidal category of A-graded complex vector spaces. Then:
- Equivalence classes of braided structures on are classified by elements of .
- also classifies -valued quadratic forms on A.
- , where the right side is "Eilenberg-MacLane abelian group cohomology" (defined in MacLane’s 1950 ICM address).
There is an additional neat interpretation involving double loop maps and multiplicative torsors on A, but I don’t need that level of sophistication for the near future.