A strict -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 -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.
A strict -category, often called simply a 3-category, is a category enriched over , the cartesian monoidal category of strict -categories. Similarly, a strict 3-groupoid? is a groupoid enriched over strict 2-groupoids.
These are also called globular strict -categories and -groupoids, to emphasise the underlying geometry.
A strict -category is the same as a strict omega-category which is trivial in degree .
This is to be contrasted with a weak -category called a tricategory and a semistrict -category called a Gray-category.
has the same underlying category as the symmetric monoidal category Gray. However, a category enriched over , a Gray-category, is more general than a strict -category.