n-category

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In mathematics, n-categories are a high-order generalization of the notion of category. An n-category is an algebraic structure consisting of objects, morphisms between objects, 2-morphisms between morphisms, and so on up to n-morphisms, and having various ways of composing these morphisms. Weak n-categories have all rules governing the composition of j-morphisms hold only up to equivalence. An n-morphism is an equivalence if it is invertible, while a j-morphism for j < n is an equivalence if it is invertible up to a (j+1)-morphism that is an equivalence. A 0-category is just a set, while a 1-category is just a category. n-Categories for large n play an important role in topological quantum field theory because they provide avoidance in mistaking isomorphism for equality.^[1]

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 functors. 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.

[edit] Definition

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.

An n-category is then an object of n-Cat. (Note that many other definitions have been proposed.)