nLab
1-category

Context

Category theory

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

In the context of higher category theory one sometimes needs, for emphasis, to say 1-category for category.

Details

Fix a meaning of \infty-category, however weak or strict you wish. Then a 11-category is an \infty-category such that every 2-morphism is an equivalence and all parallel pairs of j-morphisms are equivalent for j2j \geq 2. Thus, up to equivalence, there is no point in mentioning anything beyond 11-morphisms, except whether two given parallel 11-morphisms are equivalent.

If you rephrase equivalence of 11-morphisms as equality, which gives the same result up to equivalence, then all that is left in this definition is a category. Thus one may also say that a 11-category is simply a category.

The point of all this is simply to fill in the general concept of nn-category; nobody thinks of 11-categories as a concept in their own right except simply as categories.

The notions of 11-groupoid and 11-poset are defined on the same basis.

Revised on March 10, 2012 12:53:11 by Urs Schreiber (89.204.154.201)