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

Commutative diagram for associativity isomorphism

Applications

In tensor categories

A tensor category, or monoidal category, is a semi-groupal category with an identity object, $I$, such that $I\otimes A\cong A$ and $A\otimes I\cong A$. modular tensor categories have many applications in physics, especially in the field of topological quantum field theories.