# nLab (n,1)-category

Contents

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

#### Higher category theory

higher category theory

# Contents

## Idea

The special case of an (n,r)-category for $r = 1$.

## Definition

An $(n,1)$-category, is an $n$-category $C$ that is locally $(n-1)$-groupoidal; that is, for any objects $x$ and $y$, the $(n-1)$-category $C(x,y)$ is an $(n-1)$-groupoid.

Equivalently it is an $(\infty,1)$-category for which the mapping spaces are all $(n-1)$-truncated.

## Special cases:

• A $(1,1)$-category is the same as a $1$-category, which is an ordinary category.
• A $(2,1)$-category is a locally groupoidal $2$-category.
• An $(\infty,1)$-category can be understood as a quasi-category or in many other ways.

## Extra stuff, structure, property

• An $(n,1)$-category with the analogous properties of a topos is an (n,1)-topos.

## Examples

The canonical example of an $(n+1,1)$-category is nGrpd.

In Section 11 of

the author describes a presentation of $(n,1)$-categories by a left Bousfield localization of the model structure presenting complete Segal spaces.