nLab braided monoidal groupoid

Idea

A braided monoidal groupoid is a braided monoidal category whose underlying category happens to be a groupoid (hence all whose morphisms are isomorphisms.)

Equivalently: A monoidal groupoid with braiding that satisfies the hexagon identities,

Equivalently: A 1-truncated $E_2$-space.

Definitions

A braided monoidal groupoid is a monoidal groupoid $G$ with a natural unitary morphism $\beta_{A,B} : A \otimes B \cong^\dagger B \otimes A$ such that for all objects $A:G$, $B:G$, and $C:G$,

$$\alpha_{B,C,A} \circ \beta_{A, B \otimes C} \circ \alpha_{A,B,C} = (id_B \otimes \beta_{A,C}) \circ \alpha_{B,A,C} \circ (\beta_{A,B}\otimes id_C)$$

and

$$\alpha^{-1}_{C,A,B} \circ \beta_{A \otimes B, C} \circ \alpha^{-1}_{A,B,C} = (\beta_{A,C} \otimes id_B) \circ \alpha^{-1}_{A,C,B} \circ (id_A \otimes \beta_{B,C}$$

See also

• commutative monoid
• En-algebra
• groupoid
• monoidal groupoid
• braided 2-group
• symmetric monoidal groupoid
• k-tuply monoidal n-groupoid
• braided monoidal category