Weak n-category
In category theory, a weak n-category is a generalization of the notion of strict n-category where composition and identities are not strictly associative and unital, but only associative and unital up to . This generalisation only becomes noticeable at dimensions two and above where weak 2-, 3- and 4-categories are typically referred to as bicategories, tricategories, and tetracategories. The subject of weak n-categories is an area of ongoing research.
History[]
There is currently[when?] much work to determine what the coherence laws for weak n-categories should be. Weak n-categories have become the main object of study in higher category theory. There are basically two classes of theories: those in which the higher cells and higher compositions are realized algebraically (most remarkably 's theory of weak higher categories) and those in which more topological models are used (e.g. a higher category as a simplicial set satisfying some universality properties).
In a terminology due to John Baez and James Dolan, a (n, k)-category is a weak n-category, such that all h-cells for h > k are invertible. Some of the formalism for (n, k)-categories are much simpler than those for general n-categories. In particular, several technically accessible formalisms of (infinity, 1)-categories are now known. Now the most popular such formalism centers on a notion of quasi-category, other approaches include a properly understood theory of simplicially enriched categories and the approach via Segal categories; a class of examples of stable (infinity, 1)-categories can be modeled (in the case of characteristics zero) also via pretriangulated of Maxim Kontsevich. Quillen model categories are viewed as a of an (infinity, 1)-category; however not all (infinity, 1)-categories can be presented via model categories.
See also[]
- Bicategory
- Tricategory
- Tetracategory
- Infinity category
- Opetope
- Stabilization hypothesis
External links[]
- n-Categories – Sketch of a Definition by John Baez
- Lectures on n-Categories and Cohomology by John Baez
- Tom Leinster, Higher operads, higher categories, math.CT/0305049
- Simpson, Carlos (2012). Homotopy theory of higher categories. New Mathematical Monographs. Vol. 19. Cambridge: Cambridge University Press. arXiv:1001.4071. Bibcode:2010arXiv1001.4071S. MR 2883823.
- Jacob Lurie, Higher topos theory, math.CT/0608040, published version: pdf
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