nLab strict 3-category

Idea

• A strict $3$-category is a directed 3-graph equipped with a composition operation on adjacent cells (of all levels) which is strictly unital and associative.

• The concept of a strict $3$-category is the simplest generalization of a category to a 3-category. It is the one-step categorification of the concept of a strict 2-category.

Definition

A strict $3$-category, often called simply a 3-category, is a category enriched over $\Str2\Cat$, the cartesian monoidal category of strict $2$-categories. Similarly, a strict 3-groupoid? is a groupoid enriched over strict 2-groupoids.

These are also called globular strict $3$-categories and $3$-groupoids, to emphasise the underlying geometry.

Remarks

• A strict $3$-category is the same as a strict omega-category which is trivial in degree $j \geq 4$.

• This is to be contrasted with a weak $3$-category called a tricategory and a semistrict $3$-category called a Gray-category.

• $\Str\Cat$ has the same underlying category as the symmetric monoidal category Gray. However, a category enriched over $\Grey$, a Gray-category, is more general than a strict $3$-category.