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braided 3-group

Context

Group Theory

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

Contents

Definition

A braided 3-group is a braided ∞-group which is a 3-group. For GG a 3-group, a braiding on it is the following equivalent structure

  1. the structure of a 2-group on the delooping BG\mathbf{B}G;

  2. a doudle delooping B 2G\mathbf{B}^2 G;

  3. a lift of tha A-∞=E-1-algebra structure on GG to an E-2 algebra structure.

Examples

For RR a commutative ring, and Alg R2Vect RAlg_R \simeq 2 Vect_R the braided monoidal 2-category of RR-algebras, bimodules and bimodule homomorphism, the maximal 3-group

Br(R)Core(Alg R) \mathbf{Br}(R) \hookrightarrow Core(Alg_R)

inside is a braided 3-group. Its homotopy groups are the Brauer group, the Picard group and the group of units of RR. See at Brauer group – Relation to category of modules for more on this.

Revised on December 12, 2012 16:47:33 by Urs Schreiber (71.195.68.239)