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Incategory theory in mathematics, aweakn-category is a generalization of the notion ofstrictn-category where composition and identities are not strictly associative and unital, but only associative and unitalup tocoherent equivalence. This generalisation only becomes noticeable at dimensions two and above where weak 2-, 3- and 4-categories are typically referred to asbicategories,tricategories, andtetracategories.

There is much work to determine what the coherence laws for weakn-categories should be. Weakn-categories have become the main object of study inhigher category theory. There are basically two classes of theories: those in which the higher cells and higher compositions are realized algebraically (most remarkablyMichael Batanin's theory of weak higher categories) and those in which more topological models are used (e.g. a higher category as asimplicial set satisfying some universality properties).

In a terminology due toJohn Baez and James Dolan, a(n,k)-category is a weakn-category, such that allh-cells forh >k are invertible. Some of the formalism for(n,k)-categories are much simpler than those for generaln-categories. In particular, several technically accessible formalisms of(infinity, 1)-categories are now known. Now the most popular such formalism centers on a notion ofquasi-category, other approaches include a properly understood theory of simplicially enriched categories and the approach via Segal categories; a class of examples ofstable(infinity, 1)-categories can be modeled (in the case of characteristics zero) also via pretriangulatedA-infinity categories ofMaxim Kontsevich.Quillen model categories are viewed as apresentation of an(infinity, 1)-category; however not all(infinity, 1)-categories can be presented via model categories.

• Baez, John C. (1997). "An introduction to n-categories".Category Theory and Computer Science. Lecture Notes in Computer Science. Vol. 1290. pp. 1–33.arXiv:q-alg/9705009.doi:10.1007/BFb0026978.ISBN 978-3-540-63455-3.
• Baez, John C.; Dolan, James (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.
• Baez, John C.; Dolan, James (1998). "Categorification".arXiv:math/9802029.
• Baez, John C.; Shulman, Michael (2010). "Lectures on N-Categories and Cohomology".Towards Higher Categories. The IMA Volumes in Mathematics and its Applications. Vol. 152. pp. 1–68.arXiv:math/0608420.doi:10.1007/978-1-4419-1524-5_1.ISBN 978-1-4419-1523-8.
• Leinster, Tom (2001). "A Survey of Definitions of n-Category".arXiv:math/0107188.
• Leinster, Tom (2004).Higher Operads, Higher Categories.arXiv:math/0305049.doi:10.1017/CBO9780511525896.ISBN 978-0-521-53215-0.
• Lurie, Jacob (2009).Higher Topos Theory. Princeton University Press.arXiv:math/0608040.ISBN 978-0-691-14048-3.
• Simpson, Carlos (2012).Homotopy theory of higher categories. New Mathematical Monographs. Vol. 19. Cambridge:Cambridge University Press.arXiv:1001.4071.Bibcode:2010arXiv1001.4071S.ISBN 9781139502191.MR 2883823.

• Bicategory
• Tricategory
• Tetracategory
• Infinity category
• Opetope
• Stabilization hypothesis

• Baez, John."n-Categories – Sketch of a Definition".
• Baez, John."What n-Categories Should Be Like"(PDF).

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