nLab
3-category

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

A 33-category is any of several concepts that generalize 22-categories one step in higher category theory. The original notion is that of a globular strict 3-category, but the one most often used here is that of a tricategory. The concept generalizes to nn-categories.

Definition

Fix a meaning of \infty-category, however weak or strict you wish. Then a 33-category is an \infty-category such that every 4-morphism is an equivalence, and all parallel pairs of jj-morphisms are equivalent for j4j \geq 4. Thus, up to equivalence, there is no point in mentioning anything beyond 33-morphisms, except whether two given parallel 33-morphisms are equivalent. This definition may give a concept more general than your preferred definition of 33-category, but it will be equivalent; basically, you may have to rephrase equivalence of 33-morphisms as equality.

Specific versions

Examples

Revised on May 7, 2013 23:48:40 by Urs Schreiber (67.216.17.3)