# nLab 3-category

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

A $3$-category is any of several concepts that generalize $2$-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 $n$-categories.

## Definition

Fix a meaning of $\infty$-category, however weak or strict you wish. Then a $3$-category is an $\infty$-category such that every 4-morphism is an equivalence, and all parallel pairs of $j$-morphisms are equivalent for $j \geq 4$. Thus, up to equivalence, there is no point in mentioning anything beyond $3$-morphisms, except whether two given parallel $3$-morphisms are equivalent. This definition may give a concept more general than your preferred definition of $3$-category, but it will be equivalent; basically, you may have to rephrase equivalence of $3$-morphisms as equality.

## Examples

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