nLab cubical category

The term cubical category has at least two common meanings. Thus, to avoid ambiguity, it is perhaps better to avoid it entirely and use an equivalent, unambiguous term for the particular meaning one has in mind.

• The cubical category $\Box$ is the domain category for the presheaf category of cubical sets. To avoid ambiguity, $\Box$ may also be called the cube category (which see, for its definition) or the cubical indexing category.

• A cubical category is a cubical object in Cat (that is, a functor from $\Box^{op}$ to $Cat$), just like a cubical set is a cubical object in Set, a cubical space is a cubical object in Top, a cubical abelian group is a cubical object in Ab, and so on. To avoid ambiguity, cubical objects in Cat? may be called exactly that.

A third use of the adjective “cubical” with a type of category is the following.

• A cubical ∞-category is a cubical set equipped with operations giving a way to compose cubes with each other, in the same way that an ordinary ∞-category is a globular set equipped with composition operations.