nLab n-category



Higher category theory

higher category theory

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In broad generality, under nn-categories one understands higher categories with non-trivial higher morphisms up to dimension n{}n \in \mathbb{N} \sqcup \{\infty\}. For n=1n = 1 these are the ordinary categories (or 1-categories, for emphasis) of category theory, for n=2n = 2 they are the 2-categories (traditionally: “bicategories”) and/or double categories of 2-category theory, for n=3n = 3 one also speaks of tricategories, etc. The case n=n = \infty would broadly refer to the most general notion of \infty -categories where there is no constraint on the dimensionality of higher morphisms.

Especially as nn increases, there is a plethora of different definitions of nn-categories, some differing in generality others different-looking but secretly equivalent. A (woefully incomplete) list is given below, with pointers to dedicated entries. Part of the subject of higher category theory is to understand, organize, systematize and, last not least, apply these definitions. (It is the “nn” in “nn-category” that gives the nLab its name.)

Often it is important to consider the more fine-grained notion of ( k , n ) (k,n) -categories, where now nn refers only to the maximal dimension of non-invertible higher morphisms (non-equivalences) while the invertible higher morphisms are allowed to range in dimension up to k{}k \in \mathbb{N} \sqcup \{\infty\}.

Particularly important here is the limiting case (k=k = \infty) of ( , n ) (\infty,n) -categories: Since ( , 0 ) (\infty,0) -categories aka \infty -groupoids are the bare homotopy types of “spaces” as considered in homotopy theory, ( , n ) (\infty,n) -categories are the natural notion of nn-categories internal to homotopy theory (also: internal to homotopy type theory, up some technical issues) where the notoriously intricate coherence laws of higher category theory typically have a particularly natural (if maybe non-explicit) formulation. Therefore many authors these days use “\infty-category” as a pseudonym for ( , 1 ) (\infty,1) -category and some use “nn-category” as a tacit pseudonym for ( , n ) (\infty,n) -categories.

In this homotopy theoretic-perspective on nn-categories they are naturally understood as (models for) directed spaces (with an nn-fold notion of “direction”) and in principle the topics of directed homotopy theory (and directed homotopy type theory) ought to be thought of as essentially synonymous to ( , n ) (\infty,n) -category theory, though a genuine perspective of directed homotopy/directed homotopy type theory is still in its infancy.

Conversely, for concrete computations it is at times convenient to have less flexible and more rigid definitions or at least presentations/models of nn-categories. In as far as these are still suitably equivalent to the general notion one speaks of semi-strict n-categories and much of higher category theory revolves around identifying and comparing such rigidifications. On the other hand, perfectly strict n n -categories are typically too rigid to play more than a niche role in higher category theory.

Finally, it is sometimes useful to subsume degenerate cases in the general pattern of nn-categories, such as 0-categories (sets), (0,1)-categories (posets) and even (-1)-categories (truth values) and (-2)-categories (the singleton category).

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.”


  • 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.


See also the references at higher category theory and especially at (infinity,n)-category.

Exposition of the original algebraic notions of nn-categories:

Last revised on February 13, 2024 at 11:47:13. See the history of this page for a list of all contributions to it.