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.

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

Further reading

  • Todd Trimble, Notes on Tetracategories, October 2006,
  • . ncatlab.org.
  • . ncatlab.org.
  • Buhné, Lukas (2015). (Thesis). Staats- und Universitätsbibliothek Hamburg Carl von Ossietzky. - Theorem 2.12 (The Yoneda lemma for tricategories).
  • in Japanese