N-category
Encyclopedia
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:
over the category n-Cat.
An n-category is then an object of 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 higher category theory
, where several types of n-categories are studied. The necessity of weakening the definition of an 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, n-categories are a high-order generalization of the notion of category
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
. 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 functionsCategory of setsIn the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
, - the category (
Enriched category
In category theory and its applications to mathematics, enriched category is a generalization of category that abstracts the set of morphisms associated with every pair of objects to an opaque object in some fixed monoidal category of "hom-objects" and then defines composition and identity solely...
over the category n-Cat.
An n-category is then an object of n-Cat.
The monoidal
Monoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
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
Product (category theory)
In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...
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
Category of small categories
In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories...
and functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
s.
n-categories have given rise to higher category theory
Higher category theory
Higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities.- Strict higher categories :...
, where several types of n-categories are studied. The necessity of weakening the definition of an n-category for homotopic
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
purposes has led to the definition of weak n-categories
Weak n-category
In category theory, weak n-categories are a generalization of the notion of n-category where composition is not strictly associative but only associative up to coherent equivalence. There is currently much work to determine what the coherence laws should be for those. Weak n-categories have become...
. For distinction, the n-categories as defined above are called strict.
See also
- 2-category2-categoryIn category theory, a 2-category is a category with "morphisms between morphisms"; that is, where each hom set itself carries the structure of a category...
- n-category numberN-category numberIn mathematics, the n-category number of a mathematician is a humorous construct invented by Dan Freed, intended to measure the capacity of that mathematician to stomach the use of higher categories. It is defined as the largest number n such that he or she can think about n-categories for a half...
- Weak n-categoryWeak n-categoryIn category theory, weak n-categories are a generalization of the notion of n-category where composition is not strictly associative but only associative up to coherent equivalence. There is currently much work to determine what the coherence laws should be for those. Weak n-categories have become...
- infinity category