2groupoid

A 2-groupoid is a mathematical structure that generalizes the concept of a groupoid to the 2-dimensional setting. In a 2-groupoid, there are two types of morphisms: 0-morphisms, which are the objects, and 1-morphisms, which are the arrows between objects. Additionally, there are 2-morphisms, which are the arrows between 1-morphisms. This structure allows for the composition of morphisms in a way that respects the associativity and unitality properties, similar to how it is done in a category.

The composition of 1-morphisms in a 2-groupoid is not necessarily associative, but it is associative up to a 2-morphism, which is called a coherence condition. This means that the composition of 1-morphisms is associative only when considered up to the action of 2-morphisms. Similarly, the identity 1-morphisms are not necessarily identity elements for composition, but they are identity elements up to a 2-morphism.

2-groupoids are closely related to other mathematical structures, such as crossed modules and double groupoids. They have applications in various areas of mathematics, including homotopy theory, where they are used to study higher-dimensional homotopy types, and in category theory, where they provide a framework for understanding the structure of higher-dimensional categories.