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symmetric 2-group

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Group Theory

(,1)(\infty,1)-Category theory

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Definition

Definition

A symmetric 2-group is a 2-group equipped with the following equivalent structure:

  1. Regarded as a monoidal category, GG is a symmetric monoidal category.

  2. Regarded as a 1-truncated ∞-group it has the structure of an abelian ∞-group.

  3. The delooping 2-groupoid BG\mathbf{B}G is a braided 3-group.

  4. The double delooping 3-groupoid B 2G\mathbf{B}^2 G is a 4-group.

  5. The triple delooping 4-groupoid B 4G\mathbf{B}^4 G exists.

  6. The A-∞ algebra/E1-algebra structure on GG refines to an E3-algebra structure.

  7. GG is a 3-tuply monoidal groupoid.

  8. GG is a groupal 3-tuply monoidal (1,0)-category.

Revised on December 13, 2012 04:23:26 by Urs Schreiber (71.195.68.239)