Associativity isomorphism: Difference between revisions

Concept from category theory

In mathematics, specifically in the field of category theory, the associativity isomorphism implements the notion of associativity with respect to monoidal products in semi-groupal (or monoidal-without-unit) categories.

A category, C {\displaystyle {\mathcal {C}}} , is called semi-groupal if it comes equipped with a functor C × C → C {\displaystyle {\mathcal {C}}\times {\mathcal {C}}\to {\mathcal {C}}} such that the pair ( A , B ) ↦ A ⊗ B {\displaystyle (A,B)\mapsto A\otimes B} for A , B ∈ ob ( C ) {\displaystyle A,B\in {\text{ob}}({\mathcal {C}})} , as well as a collection of natural isomorphisms known as the associativity isomorphisms (or “associators”).^{[1][2][full citation needed]} These isomorphisms, a X , Y , Z : X ⊗ ( Y ⊗ Z ) → ( X ⊗ Y ) ⊗ Z {\displaystyle a_{X,Y,Z}:X\otimes (Y\otimes Z)\to (X\otimes Y)\otimes Z} , are such that the following “pentagon identity” diagram commutes.

A tensor category,^{[3][full citation needed]} or monoidal category, is a semi-groupal category with an identity object, I {\displaystyle I} , such that I ⊗ A ≅ A {\displaystyle I\otimes A\cong A} and A ⊗ I ≅ A {\displaystyle A\otimes I\cong A} . modular tensor categories have many applications in physics,^{[speculation?]} especially in the field of topological quantum field theories.^{[4][unreliable source?][5][dubious – discuss]}