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For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.

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 ${\rm Cat}$, which is the category of small categories and functors is actually a $2$-category with natural transformations as its $2$-morphisms, the category $n{\rm Cat}$ 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 $n{\rm Cat}$.

So a $1$-category is just a (locally small) category.

The monoidal structure of ${\rm Set}$ 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 this monoidal structure, known as the Cartesian monoidal structure. The recursive construction of $n{\rm Cat}$ works because if a category ${\cal C}$ has finite products then the category of ${\cal C}$-enriched categories has finite products as well.

In homotopy theory the above notions are often replaced with their homotopy coherent variants, known as $(\infty,n)$-categories. These allow for a similar recursive definition where one replaces sets with a suitable notion of spaces (or $\infty$-groupoids) and interprets the enrichment in a homotopy coherent sense.

See also:

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