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.<ref>{{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| url-access=subscription }}</ref><ref>{{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 }}</ref>
== 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,<ref>{{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 }}</ref> 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 <math>f : x \to y</math> in a strict 2-category ''C'' and <math>g:a \to b</math> in ''D'', the usual product is given as <math>f \times g : (x, a) \to (y, b)</math> that factors both as <math>u = (\operatorname{id}, g) \circ (f, \operatorname{id})</math> and <math>v = (f, \operatorname{id}) \circ (\operatorname{id}, g)</math>. The Gray tensor product <math>f \otimes g</math> weakens this so that we merely have a 2-morphism from <math>u</math> to <math>v</math>.<ref>Introduction in Sjoerd E. Crans, A tensor product for Gray-categories, Theory and Applications of Categories 5 (1999), no. 2, 12–69.</ref> 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. {{uncited|date=February 2026}}
== References == {{reflist}} *{{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 |author-link=Tom Leinster | 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}}
== Further reading == * Todd Trimble, Notes on Tetracategories, October 2006, [https://math.ucr.edu/home/baez/trimble/tetracategories.html] * {{Cite web |title=Gray-category in nLab |url=https://ncatlab.org/nlab/show/Gray-category|website=ncatlab.org}} *{{Cite web |title=Strict 3-category in nLab |url=https://ncatlab.org/nlab/show/strict+3-category|website=ncatlab.org}} *{{cite thesis |last1=Buhné |first1=Lukas |title=Topics in three-dimensional descent theory |url=https://ediss.sub.uni-hamburg.de/handle/ediss/6382 |publisher=Staats- und Universitätsbibliothek Hamburg Carl von Ossietzky |language=en |date=2015}} - Theorem 2.12 (The Yoneda lemma for tricategories). * http://pantodon.jp/index.rb?body=Gray-tensor_product in Japanese
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Category:Category theory