nLab n-category



Higher category theory

higher category theory

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For nn \in \mathbb{N}, an nn-category is like

  • an nn-truncated directed space in which (kn)(k \leq n)-dimensional paths have a direction, while all higher dimensional paths are reversible and parallel higher dimensional paths are homotopic;

  • a nn-fold higher analog of what a category is to a set

nn-Categories are the main subject of higher category theory, and give the nn-Lab its name. In their modern formulation in homotopy theory they are known as (∞,n)-categories (see there for more details).

Semi-formally, nn-categories can be described as follows. An nn-category is an ∞-category such that all (n+1)(n+1)-morphisms are equivalences, and all parallel pairs of jj-morphisms are equivalent for j>nj \gt n. (One says that the \infty-category is trivial in degree greater than nn.) This is the same thing as an (n,n)(n,n)-category in the sense of (n,r)(n,r)-categories.

Up to equivalence, you may assume that all equivalent pairs of jj-morphisms for j>nj \gt n are in fact equal, and many authorities include this as a requirement. On the other hand, you can also write down a definition of nn-category from scratch (without passing through \infty-categories), and then this question never comes up. The point is that you don't talk about jj-morphisms for j>nj \gt n; you stop at nn-morphisms.

On the nnLab, the term “nn-category” usually means a weak nn-category, in which the compositions of cells obeys the usual associativity, unit, and exchange laws only up to coherent equivalence. This sort of nn-category is somewhat tricky to define; there are a number of proposals, not yet shown to be equivalent. By contrast, strict n-categories are easy to define, but are not sufficient for most examples when n3n\ge 3 (see semistrict n-category).


One also speaks of (1)(-1)-categories and (2)(-2)-categories, but these concepts are not as well behaved.

Categories of nn-categories

Just as the collection of all (small) sets is the prototypical example of a category, so the collection of all small nn-categories is the prototypical example of an (n+1)(n+1)-category.

Actually, if you define things cleverly, then you can get an (n+1)(n+1)-category of all nn-categories. If one assumes the Axiom of Universes, then there is a sequence of Grothendieck universes

U 0U 1U 2U_0 \subset U_1 \subset U_2 \subset \cdots

and we can say a set is U iU_i-small if it is an element of U iU_i. This allows us to make the following definitions:

  • Set\Set is the category of all U 0U_0-small sets;
  • Cat\Cat is the 2-category of all U 1U_1-small categories;
  • 2Cat2\Cat is the 3-category of all U 2U_2-small 2-categories;
  • etc.

This is a convenient way to settle size questions once and for all for finite nn, but it doesn't really work for \infty-categories.

For more, see the discussion at sci.logic.


Here is a list of (some of) the proposed definitions of (weak) nn-category, with references, and also a list of (some of) the comparisons that have been done.

List of definitions

Many of these definitions are actually “truncations” of definitions of (weak) ∞-categories (aka ∞-categories). Some others are truncations of a definition of (∞,n)-categories. A nice overview of (many) of these can be found in Tom Leinster’s paper “A survey of definitions of nn-category.”

Someone should add some more references!


  • All definitions produce the correct well-known notion of 1-category, up to minor inessential details.

  • Since all the common definitions of (∞,1)-category are known to be equivalent (give references!), the definitions of Street, Trimble, Tamsamani–Simpson, Joyal, Barwick, and Rezk can be said to agree for (,1)(\infty,1)-categories.

  • Julie Bergner has shown that all the notions of \infty-groupoid obtained from the common notions of (,1)(\infty,1)-category are equivalent, so the definitions of Street, Trimble, Tamsamani–Simpson, Joyal, Barwick, and Rezk can also be said to agree for (,0)(\infty,0)-categories.

  • It is known that the notions of (n,0)(n,0)-category obtained from categories, bicategories, and tricategories model all homotopy n-types for n3n\le 3. Thus, in these cases, the classical definitions can be said to agree with those listed in the previous example.

  • In Tom Leinster’s paper, proofs are sketched showing that the notion of 2-category obtained in each case looks somewhat like the notion of bicategory.

  • Nick Gurski has shown in “Nerves of bicategories as stratified simplicial sets” that Street’s definition is correct for n=2n=2 (that is, it agrees with bicategories).

  • Eugenia Cheng has shown that the opetopic definition is correct for n=2n=2 (that is, it agrees with bicategories).

  • Eugenia Cheng has more recently also shown that from any sequence of operads used for iterated enrichment in a Trimble-style definition, one can construct a Batanin–Leinster-style globular operad whose algebras are the nn-categories obtained in the Trimblean inductive manner. Not all globular operads can be obtained in this way, however, since those that arise have strict interchange.

Please add any other comparisons you are aware of!

Last revised on October 3, 2022 at 15:41:16. See the history of this page for a list of all contributions to it.