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, , 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 (or "associators"). These isomorphisms, , are such that the following "pentagon identity" diagram commutes.
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.