nLab
(infinity,n)Cat

Idea

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

• objects are (∞,n)-categories;

• morphisms are (∞,n)-functors;

• k-morphisms for $k\ge 2$ 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 $(\mathrm{\infty },n)\mathrm{Cat}$. This is what is presented by various model category structures on models for $(\mathrm{\infty },n)$-categories.

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

Properties

Automorphisms

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

This is due to (BarwickSchommer-Pries).

Related concepts

• Pos

• Set

• Grpd, ∞Grpd

• Cat, Operad

• 2Cat

• (∞,1)Cat, (∞,1)Operad

• $(\mathrm{\infty },n)\mathrm{Cat}$

References

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

category: category