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 categories.

Definition

A category,, is called semi-groupal if it comes equipped with a functor such that the pair for, as well as a collection of natural isomorphisms known as the associativity isomorphisms. These isomorphisms,, are such that the following "pentagon identity" diagram commutes.

Applications

In tensor categories

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