nLab monoidal groupoid

Idea

A monoidal category whose underlying category is a groupoid (hence all whose morphisms are invertible).

Equivalently: an $A_4$-spatial groupoid whose unitors satisfy the triangle identities;

Equivalently: a 1-truncated $A_\infty$-space/$E_1$-space.

Definitions

A monoidal groupoid is an $A_4$-spatial groupoid $G$ such that the triangle identity is satisfied for all objects $A:G$ and $B:G$:

$$\tau_{A,B}:(id_A \otimes \lambda_B) \circ \alpha_{A,I,B} = (\rho_A \otimes id_B)$$

Properties

The oidification of a monoidal groupoid is a (2,1)-category.

See also

• monoid
• A-infinity space
• En-algebra
• groupoid
• A4-spatial groupoid
• 2-group, Picard groupoid
• braided monoidal groupoid
• k-tuply monoidal n-groupoid
• ring groupoid
• 2-action?
• monoidal category, monoidal dagger category
• (2,1)-category

algebraic structure	oidification
truth value	preorder
magma	magmoid
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid?	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory