In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
braided monoidal category is a
monoidal categoryIn mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative , and an object I which is both a left and right identity for ⊗,...
C equipped with a
braiding; that is, there is a natural isomorphism
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
braided monoidal category is a
monoidal categoryIn mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative , and an object I which is both a left and right identity for ⊗,...
C equipped with a
braiding; that is, there is a natural isomorphism
for which the following hexagonal diagrams commute (here is the associativity isomorphism):
Alternatively, a braided monoidal category can be seen as a
tricategoryIn mathematics, a tricategory is a kind of structure of category theory studied in higher-dimensional category theory.Whereas a weak 2-category is said to be a bicategory [Benabou 1967], a weak 3-category is said to be a tricategory .Tetracategories are the corresponding notion in dimension four...
with one 0-cell and one 1-cell.
A
symmetric monoidal category is a braided monoidal category whose braiding satisfies for all objects
A and
B.
Symmetric monoidal categories provide a model for
linear logicLinear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter...
and linear types, much like
cartesian closed categoriesIn category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in...
do for intuitionistic and classical logic.
Properties
In a braided monoidal category, the braiding always "commutes with the units":
External links