In mathematics, especially in category theory, a 3-category is a 2-category together with 3-morphisms. It comes in at least three flavors
The coherence theorem of Gordon–Power–Street says a weak 3-category is equivalent (in some sense) to a Gray category.
Strict and weak 3-categories.
A strict 3-category is defined as a category enriched over 2Cat, the monoidal category of (small) strict 2-categories. A weak 3-category is then defined roughly by replacing the equalities in the axioms by coherent isomorphisms.
Gray tensor product.
Introduced by Gray, a Gray tensor product is a replacement of a product of 2-categories that is more convenient for higher category theory. Precisely, given a morphism formula_1 in a strict 2-category "C" and formula_2 in "D", the usual product is given as formula_3 that factors both as formula_4 and formula_5. The Gray tensor product formula_6 weakens this so that we merely have a 2-morphism from formula_7 to formula_8. Some authors require this 2-morphism to be an isomorphism, amounting to replacing lax with pseudo in the theory.
Let Gray be the monoidal category of strict 2-categories and strict 2-functors with the Gray tensor product. Then a Gray category is a category enriched over Gray.
Variants.
Tetracategories are the corresponding notion in dimension four. Dimensions beyond three are seen as increasingly significant to the relationship between knot theory and physics.