2groupoids

A 2-groupoid is a category where every morphism is an isomorphism, and where the composition of 1-morphisms, called 2-morphisms, is also associative up to a specified coherent isomorphism. In essence, it is a category where all squares commute up to a natural transformation. The formal definition involves a collection of objects and two types of morphisms, often called 0-cells and 1-cells, and 1-cells and 2-cells, respectively. There are two composition operations: one for 1-cells and one for 2-cells. The composition of 1-cells is associative, and each 1-cell has an inverse. The composition of 2-cells is also associative. Crucially, there is an interchange law relating the composition of 1-cells and 2-cells, which states that the composition of 2-cells is compatible with the composition of 1-cells. This interchange law is a defining characteristic of higher categories. 2-groupoids are a fundamental concept in higher category theory and have applications in various fields, including algebraic topology, theoretical physics, and computer science. They provide a framework for understanding structures that are "categorically" rich, extending the notions found in ordinary groupoids. The concept generalizes groupoids, which are categories where every morphism is an isomorphism, to a higher dimensional setting.