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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 15th 2013
   • (edited Sep 15th 2013)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAt [[super algebra]] it says >The crucial super-grading rule (the “Koszul sign rule”) $$ a \otimes b = (-1)^{deg(a) deg(b)} b \otimes a $$ >in the symmetric monoidal category of Z-graded vector spaces is induced from the subcategory which is the abelian 2-group of metric graded lines. This in turn is the free abelian 2-group (groupal symmetric monoidal category) on a single generator. What's the general $n$-story here? In this case we found a symmetric monoidal category whose [[Picard 2-group]] is the free abelian 2-group on one generator. How would I chose the former? For $n = 1$, which monoidal cateory do I chose with Picard group equal to $\mathbb{Z}$? For $n = \infty$, what has Picard $\infty$-group the sphere spectrum? And then what are the equivalents of the Koszul sign rule. Presumably, for $n = 1$, it's just ordinary commutative multiplication. I guess some of this is linked to the Baez-Dolan predication from [Categorification](http://arxiv.org/abs/math/9802029) that if $G_{n, k}$ denotes the ‘free k-tuply groupal n-groupoid on one object’, then $$ G_{n, k} = \Pi_n(\Omega^k S^k). $$

   At super algebra it says

   The crucial super-grading rule (the “Koszul sign rule”)

   in the symmetric monoidal category of Z-graded vector spaces is induced from the subcategory which is the abelian 2-group of metric graded lines. This in turn is the free abelian 2-group (groupal symmetric monoidal category) on a single generator.

   What’s the general -story here?

   In this case we found a symmetric monoidal category whose Picard 2-group is the free abelian 2-group on one generator. How would I chose the former?

   For , which monoidal cateory do I chose with Picard group equal to ? For , what has Picard -group the sphere spectrum?

   And then what are the equivalents of the Koszul sign rule. Presumably, for , it’s just ordinary commutative multiplication.

   I guess some of this is linked to the Baez-Dolan predication from Categorification that if denotes the ‘free k-tuply groupal n-groupoid on one object’, then