Higher category theory
Encyclopedia
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.
theory:
0-categories are sets, and (n+1)-categories are categories enriched over the monoidal category of n-categories (with the monoidal structure given by finite products). This construction is well defined, as shown in the article on n-categories.
This concept introduces higher arrows, higher compositions and higher identities, which must well behave together. For example, the category of small categories is in fact a 2-category, with natural transformations as second degree arrows. However this concept is too strict for some purposes (for example, homotopy theory), where "weak" structures arise in the form of higher categories.
is the composition of paths
, which is associative only up to homotopy
. These isomorphisms must well behave between hom-sets and expressing this is the difficulty in the definition of weak n-categories. Weak 2-categories, also called bicategories
, were the first to be defined explicitly. A particularity of these is that a bicategory with one object is exactly a monoidal category
, so that bicategories can be said to be "monoidal categories with many objects." Weak 3-categories, also called tricategories
, and higher-level generalizations are increasingly harder to define explicitly. Several definitions have been given, and telling when they are equivalent, and in what sense, has become a new object of study in category theory.
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...
at a higher order, which means that some equalities are replaced by explicit arrows
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
in order to be able to explicitly study the structure behind those equalities.
Strict higher categories
N-categories are defined inductively using the enriched categoryEnriched 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...
theory:
0-categories are sets, and (n+1)-categories are categories enriched over the monoidal category of n-categories (with the monoidal structure given by finite products). This construction is well defined, as shown in the article on n-categories.
This concept introduces higher arrows, higher compositions and higher identities, which must well behave together. For example, the category of small categories is in fact a 2-category, with natural transformations as second degree arrows. However this concept is too strict for some purposes (for example, homotopy theory), where "weak" structures arise in the form of higher categories.
Weak higher categories
In weak n-categories, the associativity and identity conditions are no longer strict (that is, they are not given by equalities), but rather are satisfied up to an isomorphism of the next level. An example in topologyTopology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
is the composition of paths
Path (topology)
In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
, which is associative only up to homotopy
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...
. These isomorphisms must well behave between hom-sets and expressing this is the difficulty in the definition of weak n-categories. Weak 2-categories, also called bicategories
Bicategory
In mathematics, a bicategory is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not associative, but only associative up to an isomorphism. The notion was introduced in 1967 by Jean Bénabou.Formally, a bicategory B...
, were the first to be defined explicitly. A particularity of these is that a bicategory with one object is exactly a monoidal category
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...
, so that bicategories can be said to be "monoidal categories with many objects." Weak 3-categories, also called tricategories
Tricategory
In mathematics, a tricategory is a kind of structure of category theory studied in higher-dimensional category theory.Whereas a weak 2-category is said to be a bicategory [Benabou 1967], a weak 3-category is said to be a tricategory .Tetracategories are the corresponding notion in dimension four...
, and higher-level generalizations are increasingly harder to define explicitly. Several definitions have been given, and telling when they are equivalent, and in what sense, has become a new object of study in category theory.
Quasicategories
Weak Kan complexes, or quasi-categories, are semisimplicial complexes satisfying a weak version of the Kan condition. Joyal showed that they are a good foundation for higher category theory. Recently the theory has been systematized further by Jacob Lurie who simply call them infinity categories, though the latter term is also a generic term for all models of (infinity,k) categories for any k.Simplicially enriched category
Simplicially enriched categories, or simplicial categories, are categories enriched over simplicial sets. However, when we look at them as a model for (infinity,1)-categories, then many categorical notions, say limits do not agree with the corresponding notions in the sense of enriched categories. The same for other enriched models like topologically enriched categories.Topologically enriched categories
Topologically enriched categories (sometimes simply topological categories) are categories enriched over some convenient category of topological spaces, e.g. the category of compactly generated Hausdorff topological spaces.Segal categories
These are models of higher categories introduced by Hirschowitz and Simpson in 1988, partly inspired by results of Graeme Segal in 1974.External links
- John Baez Tale of n-Categories
- The n-Category Cafe - a group blog devoted to higher category theory.