nLab braided 2-group

Definition

Definition

A 2-group $G$ is braided if it is equipped with the following equivalent structure:

1. Regarded as a monoidal category, $G$ is a braided monoidal category.

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

3. The double delooping 3-groupoid $\mathbf{B}^2 G$ exists.

4. The groupal A-∞ algebra/E1-algebra structure on $G$ refines to an E2-algebra structure.

5. $G$ is a doubly groupal groupoid.

6. $G$ is a groupal doubly monoidal (1,0)-category.

Related concepts

• braided monoidal groupoid

• k-tuply groupal n-groupoid

• symmetric 2-group

• braided ∞-group

• sylleptic 3-group

References

Under the name “braided gr-categories” or “braided cat-groups” and thought of as a sub-class of braided monoidal categories, the notion of braided 2-groups is considered in:

• André Joyal, Ross Street, Section 3 of: Braided tensor categories, Adv. Math. 102 (1993), 20–78 (doi:10.1006/aima.1993.1055)

• M. Bullejos, Pilar Carrasco, Antonio M. Cegarra, Cohomology with coefficients in symmetric cat-groups. An extension of Eilenberg–MacLane’s classification theorem, Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 163 - 189 (doi:10.1017/S0305004100071498)

• Antonio R. Garzón, Jesus G Miranda, Sec. 1 of: Homotopy theory for (braided) cat-groups, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 38 (1997) no. 2, pp. 99-139 (numdam:CTGDC_1997__38_2_99_0)

• Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, Appendix E of: On braided fusion categories I, Selecta Mathematica. New Series 16 (2010), no. 1, 1–119 (doi:10.1007/s00029-010-0017-z)

As a special case of k-tuply groupal n-groupoids:

• Antonio R. Garzón, Jesus G. Miranda, Section 6 of: Serre homotopy theory in subcategories of simplicial groups, Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123 (doi:10.1016/S0022-4049(98)00143-1)

In the generality of braided ∞-group stacks the notion appears in

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Def. 4.28 of: Principal ∞-bundles – General theory, Journal of Homotopy and Related Structures, Volume 10, Issue 4 (2015), pages 749-801 (doi:10.1007/s40062-014-0083-6, arXiv:1207.0248)

A discussion of ∞-group extensions by braided 2-groups is in

• Evan Jenkins, Extensions of groups by braided 2-groups (arXiv:1106.0772)