3morphism

A 3-morphism is a concept originating in category theory, specifically in the context of higher category theory. In a standard category, a morphism represents a "arrow" between objects. In a 2-category, there are objects, 1-morphisms between objects, and 2-morphisms between 1-morphisms. Extending this idea, a 3-category introduces 3-morphisms, which are "arrows" between 2-morphisms.

More formally, a 3-category can be thought of as a category whose objects are categories, whose morphisms are functors between these categories, and whose 2-morphisms are natural transformations between these functors. The 3-morphisms then become "natural transformations between natural transformations." This hierarchical structure allows for the formalization of increasingly complex mathematical structures and relationships.

The study of 3-morphisms and higher categories is an active area of research in abstract mathematics. It finds applications in fields such as algebraic topology, mathematical physics, and computer science, particularly in areas involving complex systems and higher-dimensional structures. While the concept can be abstract, it provides a powerful framework for understanding and organizing intricate mathematical constructions. The definition of a 3-morphism precisely captures the idea of a transformation at the level of transformations themselves.