nLab n-category

Context

Higher category theory

higher category theory

Contents

Idea

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

• a directed space in which $(k \leq n)$-dimensional paths have a direction, while all higher dimensional paths may be reversed;

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

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

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

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

On the $n$Lab, the term “$n$-category” usually means a weak $n$-category, in which the compositions of cells obeys the usual associativity, unit, and exchange laws only up to coherent equivalence. This sort of $n$-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 $n\ge 3$ (see semistrict n-category).

Examples

• A $0$-category is a set.

• A $1$-category is an ordinary category.

• A $2$-category is (depending on how strict was your initial notion of $\infty$-category) either a strict 2-category or a bicategory.

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

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 ω-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.”

Someone should add some more references!

• 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?PowerStreet), 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 ω-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.

• BaezDolan definition: an $n$-category is an opetopic set having enough $n$-universal fillers. Alternate definitions of opetopes (aka multitopes) have been given by HermidaMakkaiPower 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.

• BataninLeinster 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 ω-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.

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.