nLab (infinity,n)Cat

Idea

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

• objects are (∞,n)-categories;

• morphisms are (∞,n)-functors;

• k-morphisms for $k \geq 2$ are $(k-1)$-transfors.

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).

(See also at duality.)

Related concepts

• Pos

• Set

• Grpd, ∞Grpd

• Cat, Operad

• 2Cat

• (∞,1)Cat, (∞,1)Operad

• $(\infty,n)Cat$

References

• Clark Barwick, Chris Schommer-Pries, On the unicity of the homotopy theory of higher categories (pdf)

category: category