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An ordinary category has objects and morphisms. A 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 known as Cat, which is the category 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 (locally small) 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.

While this concept is too strict for some purposes in for example, homotopy theory, where "weak" structures arise in the form of higher categories,[1] strict cubical higher homotopy groupoids have also arisen as giving a new foundation for algebraic topology on the border between homology and homotopy theory, see the book "Nonabelian algebraic topology" referenced below.

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