nLab k-morphism

Redirected from "n-morphisms".
Higher morphisms (-morphisms)

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Higher morphisms (kk-morphisms)

Idea

In an nn-category, or most generally an \infty-category, there are many levels of morphism, parametrised by natural numbers. Those at level kk are called kk-morphisms or kk-cells.

Definition

All notions of higher category have kk-morphisms, but the shapes may depend on the model (or theory) employed.

For a simplicially based geometric model of higher categories, i.e., simplicial sets subject to some filler conditions, the kk-morphisms are literally kk-cells in the sense of a simplicial set. This applies for example to quasi-categories, weak nn-categories in the sense of Street, and the weak complicial sets of Verity. In other geometric models, based not on simplices but on other shapes such as opetopes (Baez-Dolan), multitopes (Hermida-Makkai-Power), or nn-disks (Joyal), a higher category is a presheaf

X:Shapes opSetX: Shapes^{op} \to Set

again subject to some filler conditions, and in each case kk-morphisms are elements of X(σ)X(\sigma) where σ\sigma is a shape of dimension kk. Still other shapes (e.g., cubes) are possible (see also n-fold category).

Many notions of algebraic higher category, such as those due to Batanin, Leinster, Penon, and Trimble, are algebras over certain monads acting on globular sets (such as those induced by globular operads), so that each higher category XX has an underlying globular set U(X)U(X). In that case, the kk-morphisms are the kk-cells of U(X)U(X). In such globularly based definitions, every kk-morphism ff has a (k1)(k-1)-morphism σf\sigma f as its source and a (k1)(k-1)-morphism τf\tau f as its target, and the source (k2)(k-2)-morphisms σσf\sigma \sigma f and στf\sigma \tau f must be the same, as must the target (k2)(k-2)-morphisms τσf\tau \sigma f and ττf\tau \tau f.

A 11-morphism may simply be called a morphism; a 00-morphism is an object.

Last revised on October 17, 2024 at 07:02:37. See the history of this page for a list of all contributions to it.