Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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”)

   $$a \otimes b = (-1)^{deg(a) deg(b)} b \otimes a$$

   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 that if $G_{n, k}$ denotes the ‘free k-tuply groupal n-groupoid on one object’, then

   $$G_{n, k} = \Pi_n(\Omega^k S^k).$$