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