nLab
symmetric 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
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
monoid theory in algebra :
monoid , infinity-monoid
monoid object , monoid object in an (infinity,1)-category
Mon , CMon
monoid homomorphism
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
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 braided monoidal groupoid whose dagger adjoint of the braiding is the opposite braiding ,
Equivalently, a 1-truncated
E 3
E_3
-space or
E ∞
E_\infty
-space .
Definitions
A symmetric monoidal groupoid is a braided monoidal groupoid G G such that for all objects A A and B B , the dagger adjoint or 2-sided inverse of the braiding β A , B : A ⊗ B ≅ † B ⊗ A \beta_{A,B} : A \otimes B \cong^\dagger B \otimes A is the braiding β B , A : B ⊗ A ≅ † A ⊗ B \beta_{B,A} : B \otimes A \cong^\dagger A \otimes B
β A , B † = β B , A \beta_{A,B}^\dagger = \beta_{B,A}
See also
Last revised on May 16, 2022 at 11:34:22.
See the history of this page for a list of all contributions to it.