Is there a difference between a Picard stack and a stack of symmetric monoidal categories?

It seems to me that Picard stacks as defined by Deligne and used in algebraic geometry are stacks of symmetric monoidal categories. Am I missing something? Is there a difference between the two notions? The Picard stack I am interested in shows up in symplectic toric geometry (arXiv:0908.2783v2[math.SG] )

flag	asked Jul 26 at 14:29 Eugene Lerman 575●6

The difference is obviously not in the stacky part, bur in the categorical part. A Picard category is a symmetric monomial category where all morphisms are invertible. – Fernando Muro Jul 26 at 14:35

Thanks. So a Picard category is a symmetric monoidal groupoid? – Eugene Lerman Jul 26 at 15:43

I think the point is that all objects are invertible (with respect to the monoidal structure). – Konrad Waldorf Jul 26 at 18:43

@Eugene: Your question has already been answered on MO: mathoverflow.net/questions/24912/… – Dmitri Pavlov Jul 26 at 22:18

In modern terms, a Picard category is some flavour of 'abelian' 2-group. – David Roberts Jul 26 at 22:25

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