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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] )

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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 '12 at 14:35
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Thanks. So a Picard category is a symmetric monoidal groupoid? –  Eugene Lerman Jul 26 '12 at 15:43
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I think the point is that all objects are invertible (with respect to the monoidal structure). –  Konrad Waldorf Jul 26 '12 at 18:43
    
@Eugene: Your question has already been answered on MO: mathoverflow.net/questions/24912/… –  Dmitri Pavlov Jul 26 '12 at 22:18
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In modern terms, a Picard category is some flavour of 'abelian' 2-group. –  David Roberts Jul 26 '12 at 22:25

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