n-category

In mathematics, n-categories are a high-order generalization of the notion of category.

An ordinary category has objects and morphisms. An 2-category generalizes this by also including 2-morphisms between the 1-morphisms. Continuing this up to n-morphisms between (n-1)-morphisms gives an n-category.

Just as the category Cat of small categories and functors is actually a 2-category with natural transformations as its 2-morphisms, the category n-Cat of (small) n-categories is actually an n+1-category.

An n-category is defined by induction on n by:

• A 0-category is a set,
• An (n+1)-category is a category enriched over the category n-Cat.

So a 1-category is just a category.

The monoidal structure of Set is the one given by the cartesian product as tensor and a singleton as unit. In fact any category with finite products can be given a monoidal structure. The recursive construction of n-Cat works fine because if a category C has finite products, the category of C-enriched categories has finite products too.

n-categories have given rise to higher category theory, where several types of n-categories are studied. The necessity of weakening the definition of an n-category for homotopic purposes has led to the definition of weak n-categories. For distinction, the n-categories as defined above are called strict.

See also

• n-category number
• infinity category

References

• Tom Leinster (2004). Higher Operads, Higher Categories. Cambridge University Press.
• Eugenia Cheng, Aaron Lauda (2004). Higher-Dimensional Categories: an illustrated guide book (PDF).