Kmorphism

Kmorphism is a term used in higher category theory to denote higher-dimensional morphisms within a k-category. It refers to the arrows or cells that connect morphisms at successive levels in a hierarchical structure of objects, morphisms, and higher cells.

In a k-category, objects are 0-morphisms; morphisms between objects are 1-morphisms; 2-morphisms are morphisms between 1-morphisms; and, inductively, k-morphisms are morphisms between (k-1)-morphisms for 2 ≤ k ≤ n. When k > 1, these higher morphisms carry their own composition laws: vertical composition, which composes along the same source and target, and horizontal composition, which composes between compatible pairs. This gives rise to a rich algebraic structure that generalizes ordinary categories.

In a 2-category, for example, 2-morphisms are natural transformations between functors, providing a concrete instance of a higher morphism. In a 3-category, 3-morphisms are morphisms between 2-morphisms, and so forth. The idea extends to larger frameworks such as (infinity, n)-categories, where higher morphisms exist up to a chosen dimension and coherence is governed by higher homotopies.

Notation and terminology vary: some authors use higher cells, k-cells, or n-cells to denote these levels, and some frameworks refer to infinite families of higher morphisms (as in (infinity, n)-categories). Kmorphism thus forms part of the language for describing coherent composition and transformations in higher categories, with applications across algebraic topology, algebraic geometry, and related fields. Because usage is not completely standardized, definitions are best checked within a given text.