Associativity isomorphism

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.

Definition

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

Commutative diagram for associativity isomorphism

Applications

In tensor categories

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