n-category

In mathematics, n-categories are a high-order generalization of the notion of category. The category of (small) n-categories n-Cat is defined by induction on n by:

• the category 0-Cat is the category Set of sets and functions,
• the category (n+1)-Cat is the category of categories enriched over the category n-Cat.

An n-category is then an object of n-Cat.

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.

In particular, the category 1-Cat is the category Cat of small categories and functors.

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

• 2-category
• n-category number
• Weak n-category
• infinity category

References

• Tom Leinster (2004). Higher Operads, Higher Categories. Cambridge University Press. http://www.maths.gla.ac.uk/~tl/book.html. 
• Eugenia Cheng, Aaron Lauda (2004). Higher-Dimensional Categories: an illustrated guide book. http://www.cheng.staff.shef.ac.uk/guidebook/guidebook-new.pdf.