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 $f:x\to y$ 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 $u=(\operatorname {id} ,g)\circ (f,\operatorname {id} )$ and $v=(f,\operatorname {id} )\circ (\operatorname {id} ,g)$ . The Gray tensor product $f\otimes g$ weakens this so that we merely have a 2-morphism from $u$ to $v$ .^[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.

References

^ Gordon, R.; Power, A. J.; Street, Ross (1995). "Coherence for tricategories". Memoirs of the American Mathematical Society. 117 (558). doi:10.1090/memo/0558. ISSN 0065-9266.
^ Lack, Stephen (2011). "A Quillen model structure for Gray-categories". Journal of K-Theory. 8 (2): 183–221. arXiv:1001.2366. doi:10.1017/is010008014jkt127.
^ Gray, John W. (1974). Formal Category Theory: Adjointness for 2-Categories. Lecture Notes in Mathematics. Vol. 391. doi:10.1007/BFb0061280. ISBN 978-3-540-06830-3.
^ Introduction in Sjoerd E. Crans, A tensor product for Gray-categories, Theory and Applications of Categories 5 (1999), no. 2, 12–69.

Baez, John C.; Dolan, James (10 May 1998). "Higher-Dimensional Algebra III.n-Categories and the Algebra of Opetopes". Advances in Mathematics. 135 (2): 145–206. arXiv:q-alg/9702014. 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. arXiv:math/0107188.

Strict and weak 3-categories

Gray tensor product

Variants

References

Further reading