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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, 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.