nLab symmetric 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 braided monoidal groupoid whose dagger adjoint of the braiding is the opposite braiding,

Equivalently, a 1-truncated $E_3$-space or $E_\infty$-space.

Definitions

A symmetric monoidal groupoid is a braided monoidal groupoid $G$ such that for all objects $A$ and $B$, the dagger adjoint or 2-sided inverse of the braiding $\beta_{A,B} : A \otimes B \cong^\dagger B \otimes A$ is the braiding $\beta_{B,A} : B \otimes A \cong^\dagger A \otimes B$

$$\beta_{A,B}^\dagger = \beta_{B,A}$$

See also

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