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.{{cite journal

| last1=Gordon | first1=R.

| last2=Power | first2=A. J.

| last3=Street | first3=Ross | authorlink3=Ross Street

| date=1995

| title=Coherence for tricategories

| url=http://www.ams.org/memo/0558

| journal=Memoirs of the American Mathematical Society

| language=en

| volume=117

| issue=558

| doi=10.1090/memo/0558|issn=0065-9266}}{{cite journal |doi=10.1017/is010008014jkt127 |title=A Quillen model structure for Gray-categories |date=2011 |last1=Lack |first1=Stephen |journal=Journal of K-Theory |volume=8 |issue=2 |pages=183–221 |arxiv=1001.2366 }}

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,{{cite book |url=https://doi.org/10.1007/BFb0061280 |doi=10.1007/BFb0061280 |title=Formal Category Theory: Adjointness for 2-Categories |series=Lecture Notes in Mathematics |date=1974 |volume=391 |isbn=978-3-540-06830-3|first=John W.|last=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 $u = (\operatorname{id}, g) \circ (f, \operatorname{id})$ 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 $v$ .Introduction in Sjoerd E. Crans, A tensor product for Gray-categories, Theory and Applications of Categories 5 (1999), no. 2, 12–69. 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

{{cite journal

| last=Baez |first=John C. | authorlink1=John C. Baez

| last2=Dolan | first2=James

| date=10 May 1998

| title=Higher-Dimensional Algebra III.n-Categories and the Algebra of Opetopes

| journal=Advances in Mathematics

| language=en

| volume=135

| issue=2

| pages=145–206

| doi=10.1006/aima.1997.1695 | doi-access=free

| issn=0001-8708| arxiv=q-alg/9702014

}}

{{cite journal

| last1=Leinster | first1=Tom

| date=2002

| title=A survey of definitions of n-category

| journal=Theory and Applications of Categories

| volume=10

| pages=1–70

| url=http://www.tac.mta.ca/tac/volumes/10/1/10-01abs.html

| arxiv=math/0107188}}

3-category

Strict and weak 3-categories

Gray tensor product

Variants

References

Further reading