nLab
braided monoidal groupoid
Contents
Context
Algebra
Homotopy theory
homotopy theory , (∞,1)-category theory , homotopy type theory

flavors: stable , equivariant , rational , p-adic , proper , geometric , cohesive , directed …

models: topological , simplicial , localic , …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Type theory
Monoid theory
monoid theory in algebra :

monoid , infinity-monoid

monoid object , monoid object in an (infinity,1)-category

Mon , CMon

monoid homomorphism

trivial monoid

submonoid , quotient monoid?

divisor , multiple? , quotient element?

inverse element , unit , irreducible element

ideal in a monoid

principal ideal in a monoid

commutative monoid

cancellative monoid

GCD monoid

unique factorization monoid

Bézout monoid

principal ideal monoid

group , abelian group

absorption monoid

free monoid , free commutative monoid

graphic monoid

monoid action

module over a monoid

localization of a monoid

group completion

endomorphism monoid

super commutative monoid

Categorification
categorification

Background
Contents
Examples
Category theory
Contents
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
Last revised on May 16, 2022 at 15:33:29.
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