- N-category
In
mathematics , n-categories are a high-order generalization of the notion of category. The category of (small) "n"-categories "n"-Cat is defined by induction on "n" by:
* the category 0-Cat is the category Set of sets and functions,
* the category ("n"+1)-Cat is the category of categories enriched over the category "n"-Cat.The monoidal structure of Set is the one given by the cartesian product as tensor and a singleton as unit. In fact any category with finite products can be given a monoidal structure. The recursive construction of n-Cat works fine because if a category C has finite products, the category of C-enriched categories has finite products too.
In particular, the category 1-Cat is the category Cat of small categories and
functor s.N-categories have given rise to the
higher category theory where several types of n-categories are studied. The necessity to weaken the definition of a n-category for homotopic purposes has led to the definition of weak n-categories. For distinction, the n-categories as defined above are called strict.See also
*
2-category
*Weak n-category
*n-category number References
*cite book
author = Tom Leinster
year = 2004
title = Higher Operads, Higher Categories
publisher = Cambridge University Press
url = http://www.maths.gla.ac.uk/~tl/book.html
*cite book
author = Eugenia Cheng, Aaron Lauda
year = 2004
title = Higher-Dimensional Categories: an illustrated guide book
url = http://www.math.uchicago.edu/~eugenia/guidebook/guidebook-new.pdf
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