3morphisms

A 3-morphism is a concept in higher category theory, specifically within the framework of n-categories. While a 1-morphism is a standard arrow or map between objects in a category, and a 2-morphism is a map between 1-morphisms (often called a natural transformation), a 3-morphism is a map between 2-morphisms. This forms a hierarchy where each level of structure is mapped by the next higher level.

In a strict 2-category, the composition of 1-morphisms is associative and the identity 1-morphisms act as units. A 2-morphism, also known as a 2-cell or a square, can be visualized as a transformation between two parallel 1-morphisms. Similarly, a 3-morphism, or a 3-cell, can be thought of as a transformation between two parallel 2-morphisms. This means a 3-morphism relates how two different transformations between arrows behave relative to each other.

The composition of 3-morphisms is also subject to associativity and unit laws, mirroring the structure at lower dimensions. These higher categorical structures are important for understanding more complex mathematical and physical theories, such as topological quantum field theories and certain aspects of string theory, where objects and their relationships can exist at multiple levels of abstraction. The study of 3-morphisms and higher structures is an ongoing area of research in abstract mathematics.