3-category

In mathematics, especially in category theory, a 3-category is a 2-category together with 3-morphisms. It comes in at least three flavors

• a strict 3-category,
• a semi-strict 3-category also called a Gray category,
• a weak 3-category.

The coherence theorem of Gordon–Power–Street says a weak 3-category is equivalent (in some sense) to a Gray category.^[1][2]

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,^[3] 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 ${\displaystyle f:x\to y}$ in a strict 2-category C and ${\displaystyle g:a\to b}$ in D, the usual product is given as ${\displaystyle f\times g:(x,a)\to (y,b)}$ that factors both as ${\displaystyle u=(\operatorname {id} ,g)\circ (f,\operatorname {id} )}$ and ${\displaystyle v=(f,\operatorname {id} )\circ (\operatorname {id} ,g)}$. The Gray tensor product ${\displaystyle f\otimes g}$ weakens this so that we merely have a 2-morphism from ${\displaystyle u}$ to ${\displaystyle v}$.^[4] 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.