For $n \in \mathbb{N}$, an $n$-category is like
an $n$-truncated directed space in which $(k \leq n)$-dimensional paths have a direction, while all higher dimensional paths are reversible and parallel higher dimensional paths are homotopic;
$n$-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).
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.
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
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:
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.
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.
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.”
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?–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.
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.
Please add any other comparisons you are aware of!
$n$-category