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.

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

References

• "Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes" (in en). Advances in Mathematics 135 (2): 145–206. 10 May 1998. doi:10.1006/aima.1997.1695. ISSN 0001-8708. 
• Leinster, Tom (2002). "A survey of definitions of n-category". Theory and Applications of Categories 10: 1–70. http://www.tac.mta.ca/tac/volumes/10/1/10-01abs.html. 

Further reading

• Todd Trimble, Notes on Tetracategories, October 2006, [1]
• "Gray-category in nLab". https://ncatlab.org/nlab/show/Gray-category. 
• "Strict 3-category in nLab". https://ncatlab.org/nlab/show/strict+3-category. 
• http://pantodon.jp/index.rb?body=Gray-tensor_product in Japanese