nLab n-category

Idea

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

Especially as $n$ increases, there is a plethora of different definitions of $n$-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 “$n$” in “$n$-category” that gives the nLab its name.)

Often it is important to consider the more fine-grained notion of $(k,n)$-categories, where now $n$ 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 $k \in \mathbb{N} \sqcup \{\infty\}$.

Particularly important here is the limiting case ($k = \infty$) of $(\infty,n)$-categories: Since $(\infty,0)$-categories aka $\infty$-groupoids are the bare homotopy types of “spaces” as considered in homotopy theory, $(\infty,n)$-categories are the natural notion of $n$-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 $(\infty,1)$-category and some use “$n$-category” as a tacit pseudonym for $(\infty,n)$-categories.

In this homotopy theoretic-perspective on $n$-categories they are naturally understood as (models for) directed spaces (with an $n$-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 $(\infty,n)$-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 $n$-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 $n$-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 $n$-categories, such as 0-categories (sets), (0,1)-categories (posets) and even (-1)-categories (truth values) and (-2)-categories (the singleton category).

Categories of $n$-categories

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

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

$$U_0 \subset U_1 \subset U_2 \subset \cdots$$

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

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

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

For more, see the discussion at sci.logic.

Definitions

Here is a list of (some of) the proposed definitions of (weak) $n$-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 $n$-category.”

• Classical explicit definitions of “fully weak” $n$-category exist for $n\le 4$. Weak 0-categories are sets, weak 1-categories are simply categories (due to Eilenberg and Mac Lane), weak 2-categories are bicategories (due to Benabou), weak 3-categories are tricategories (due to Gordon–Power–Street), and weak 4-categories are tetracategories (due to Todd Trimble). Going on in this way is generally admitted to be infeasible beyond $n=4$.

• Street's definition: an $n$-category is a simplicial set satisfying certain horn-filling conditions. See weak complicial set and simplicial model for weak ∞-categories. This is a truncation of a definition of $\omega$-category. It can be specialized to yield a notion of $(\infty,n)$-category. The resulting notion of $(\infty,1)$-category is a quasicategory, and the resulting notion of $\infty$-groupoid is a Kan complex.

• Baez–Dolan definition: an $n$-category is an opetopic set having enough $n$-universal fillers. Alternate definitions of opetopes (aka multitopes) have been given by Hermida–Makkai–Power and Leinster; a comparison is due to Eugenia Cheng, see these three papers. Makkai’s version can do $\omega$.

• Penon‘s definition: (someone describe this please!) Penon’s original definition turned out to be too strict (see Batanin and Cheng–Makkai) because it used reflexive globular sets, but a modification of it using globular sets is still a contender.

• Batanin–Leinster definition: an $n$-category is an $n$-globular set with an action of a suitable globular operad. This is a truncation of a definition of $\omega$-category; see Batanin ∞-category.

• Trimble-style definition: An $n$-category is a category weakly enriched over $(n-1)$-categories, where the weakness is parametrized by an operad. This definition is inductive and thus cannot do $\omega$ in an obvious way, but it has been accomplished using terminal coalgebras; see Trimble n-category. Alternately, by starting with enrichment in spaces or simplicial sets, one can obtain directly a notion of (∞,n)-category. The resulting notion of (∞,1)-category is an $A_\infty$-category.

• Tamsamani?–Simpson definition: An $n$-category is a simplicial object in $(n-1)$-categories satisfying object-discreteness and the Segal condition. This definition is inductive (it is a different way of formalizing “iterated weak enrichment”) and thus cannot do $\omega$ in an obvious way. It does have a natural extension to $(\infty,n)$-categories, and the resulting notion of (∞,1)-category reduces to a Segal category. The iterated version of this is that of Segal n-category. This notion of “weak enrichment” in a cartesian model category is studied carefully in Simpson’s book Homotopy Theory of Higher Categories.

• Moerdijk and Weiss‘s definition uses yet another way of formalizing “iterated weak enrichment,” using dendroidal sets and quasi-operad?s.

• Joyal‘s definition: An $n$-category is an $n$-cellular set satisfying horn-filling conditions. This definition can do $\omega$ by using $\omega$-cellular sets instead of $n$-cellular sets, and it can do $(\infty,n)$ by requiring different horn-filling conditions on $n$-cellular sets. The notion of (∞,1)-category one obtains in this way is a quasicategory, and the resulting notion of $\infty$-groupoid is a Kan complex. For $n\gt 1$, however, the obvious “horn-filling conditions” are not quite right; Dimitri Ara has shown how to correct them (albeit not very explicitly), obtaining a definition he calls an n-quasicategory, which form a model category Quillen equivalent to Rezk’s definition (below).

• Barwick‘s definition (popularized by Lurie in solving the Baez–Dolan cobordism hypothesis): an $(\infty,n)$-category is an $n$-fold simplicial topological space satisfying completeness and the Segal condition. See n-fold complete Segal space. An $n$-category is again defined as an $(\infty,n)$-category in which all $k$-cells are essentially unique for $k\gt n$. It is not clear whether this definition can do $\omega$. An $(\infty,1)$-category with this definition is also the same as a complete Segal space.

• Rezk‘s definition: An $(\infty,n)$-category is a simplicial $n$-cellular set satisfying fibrancy, completeness, and the Segal condition. An $n$-category can then be defined as an $(\infty,n)$-category in which all $k$-cells are essentially unique for $k\gt n$. This definition can potentially do $\omega$, although it seems not to have been written down yet. An $(\infty,1)$-category with this definition is the same as a complete Segal space. See Theta space.

• G. Maltsiniotis has apparently extracted a definition of $\infty$-groupoid from Pursuing Stacks and generalized it to a definition of $\infty$-category; see this and this.

• Simona Paoli: weakly globular n-fold categories

Comparisons

• 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 $(\infty,1)$-categories.

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

• It is known that the notions of $(n,0)$-category obtained from categories, bicategories, and tricategories model all homotopy n-types for $n\le 3$. 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 $n=2$ (that is, it agrees with bicategories).

• Eugenia Cheng has shown that the opetopic definition is correct for $n=2$ (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 $n$-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.

Related concepts

• 0-category, (0,1)-category, Set

• category

  • category object in an (∞,1)-category

• 2-category, bicategory

• 3-category, tricategory

• 4-category, tetracategory

• $n$-category

  • n-category object in an (∞,1)-category

• (n,1)-category

• (∞,1)-category

• (∞,2)-category

• (∞,n)-category

• (n,r)-category

References

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

Exposition of the original algebraic notions of $n$-categories:

• John Baez, The Tale of n-Categories

• John Baez, An Introduction to $n$-Categories, in 7th Conference on Category Theory and Computer Science, Lecture Notes in Computer Science 1290. Springer (1997)

• Eugenia Cheng, Aaron Lauda, Higher-Dimensional Categories: An Illustrated Guidebook [pdf]

• Tom Leinster, A Survey of Definitions of $n$-Category [arXiv:math.CT/0107188]

• Tom Leinster, Higher Operads, Higher Categories [arXiv:math/0305049]