# nLab (infinity,n)Cat

### Context

#### Higher category theory

higher category theory

## 1-categorical presentations

#### categories of categories

$(n+1,r+1)$-categories of (n,r)-categories

# Contents

## Idea

The (large) $(\infty,n+1)$-category $(\infty,n)Cat$ is the collection of all (small) (∞,n)-categories:

There are various presentations for this. For general $n$ see for instance this section at Theta-space. For low $n$ see the discussion at (∞,1)Cat and (∞,2)Cat?.

Often it is useful to consider just the maximal (∞,1)-category inside $(\infty,n)Cat$. This is what is presented by various model category structures on models for $(\infty,n)$-categories.

The discussion in (BarwickSchommer-Pries) shows that essentially all proposed models for $(\infty,n)Cat$ are in fact equivalent.

## Properties

### Automorphisms

The automorphism ∞-group of $(\infty,n)Cat$ is equivalent to $(\mathbb{Z}_2)^n$.

This is due to (Barwick & Schommer-Pries).

• Pos

• Set

• 2Cat

• $(\infty,n)Cat$

## References

category: category

Revised on February 27, 2014 08:31:34 by Urs Schreiber (82.113.98.143)