nLab
(infinity,n)-category

Idea

Combining the idea of $(\mathrm{\infty },1)$-category with that of $n$-category, an $(\mathrm{\infty },n)$-category is supposed to be an $\mathrm{\infty }$-category which behaves like its $k$-morphisms for $k>n$ are all equivalences. See (n,r)-category.

Idea in terms of complete Segal spaces

One definition building on that of (∞,1)-category in terms of complete Segal spaces was given in 2005 by Clark Barwick and recently put to use and popularized by Jacob Lurie in On the Classification of Topological Field Theories and in more detail in

• Jacob Lurie, $(\mathrm{\infty },2)$-Categories and the Goodwillie Calculus I (arXiv)

Due to this definition an $(\mathrm{\infty },n)$-category is an n-fold complete Segal space.

Another variant of this idea is higher Segal spaces.

For each $n\in \mathbb{N}$, there is an $(\mathrm{\infty },1)$-category $(\mathrm{\infty },n)\mathrm{Cat}$ of all $(\mathrm{\infty },n)$-categories. This is such that $(\mathrm{\infty },1)\mathrm{Cat}$ is indeed equivalent to the (∞,1)-category of (∞,1)-categories.

See def. 1.3.6 in the above reference.

Another definition also building on that of (∞,1)-category in terms of complete Segal spaces was given in early 2009 by Charles Rezk in

• Charles Rezk, A cartesian presentation of weak n-categories (arXiv)

In the above paper, they are called $(\mathrm{\infty },n)$-$\Theta$-spaces.

This definition makes use the shape categories ${\Theta }_n$ originally introduced by André Joyal; $(\mathrm{\infty },n)$-categories are defined to be the fibrant objects in an appropriate Bousfield localization of the injective model structure on simplicial presheaves on ${\Theta }_n$.

Special cases

See also

• ∞-groupoid = $(\mathrm{\infty },0)$-category
• (∞,1)-category
• (∞,2)-category
• etc …
• ∞-category = $(\mathrm{\infty },\mathrm{\infty })$-category.

In addition,

• (m,n)-categories can be obtained as particular $(\mathrm{\infty },n)$-categories whose $k$-cells are trivial for $k>m$.
• In particular, n-categories = $(n,n)$-categories can be so obtained.

Definitions

A list of known definitions of $n$-category, which should include all definitions of $(\mathrm{\infty },n)$-category as well, can be found at the page n-category. Note that some of these definitions (in particular, the “inductive” ones) only work for $n<\mathrm{\infty }$.

Examples

• One motivating example for $(\mathrm{\infty },n)$-categories is the (∞,n)-category of cobordisms which plays a central role in the formalization of the cobordism hypothesis.

• Another class of examples are (∞,n)-categories of spans.

Related concepts

• 0-category, (0,1)-category

• category

• 2-category

• 3-category

• n-category

• (∞,0)-category

• (∞,1)-category

• (∞,2)-category

• (∞,n)-category

• (n,r)-category

References

A summary of definitions and some known comparison results can be found at

• Julie Bergner, Models for $(\mathrm{\infty },n)$-categories and the cobordism hypothesis, arXiv

• Julie Bergner, Models for $(\mathrm{\infty },n)$-Categories and the Cobordism Hypothesis , in Branislav Jurčo, Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory

Axiomatic characterization is in

• Clark Barwick, Chris Schommer-Pries, On the Unicity of the Homotopy Theory of Higher Categories (pdf, unusual slides)