Higher category theory

Last updated

In mathematics, 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. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces, such as their fundamental weak ∞-groupoid.


Strict higher categories

An ordinary category has objects and morphisms, which are called 1-morphisms in the context of higher category theory. A 2-category generalizes this by also including 2-morphisms between the 1-morphisms. Continuing this up to n-morphisms between (n 1)-morphisms gives an n-category.

Just as the category known as Cat, which is the category of small categories and functors is actually a 2-category with natural transformations as its 2-morphisms, the category n-Cat of (small) n-categories is actually an (n +1)-category.

An n-category is defined by induction on n by:

So a 1-category is just a (locally small) category.

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.

While this concept is too strict for some purposes in for example, homotopy theory, where "weak" structures arise in the form of higher categories, [1] strict cubical higher homotopy groupoids have also arisen as giving a new foundation for algebraic topology on the border between homology and homotopy theory; see the article Nonabelian algebraic topology, referenced in the book below.

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 topology is the composition of paths, where the identity and association conditions hold only up to reparameterization, and hence up to homotopy, which is the 2-isomorphism for this 2-category. These n-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.


Weak Kan complexes, or quasi-categories, are simplicial sets satisfying a weak version of the Kan condition. André Joyal showed that they are a good foundation for higher category theory. Recently, in 2009, the theory has been systematized further by Jacob Lurie who simply calls them infinity categories, though the latter term is also a generic term for all models of (infinity, k) categories for any k.

Simplicially enriched categories

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 (e.g., 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 called topological categories) are categories enriched over some convenient category of topological spaces, e.g. the category of compactly generated Hausdorff spaces.

Segal categories

These are models of higher categories introduced by Hirschowitz and Simpson in 1998, [2] partly inspired by results of Graeme Segal in 1974.

See also


  1. Baez & Dolan 1998 , p. 6
  2. Hirschowitz, André; Simpson, Carlos (2001). "Descente pour les n-champs (Descent for n-stacks)". arXiv: math/9807049 .

Related Research Articles

Category theory Branch of mathematics

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows. A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.

Algebraic topology Branch of mathematics

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

The following outline is provided as an overview of and guide to category theory, the 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. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

In mathematics, the Seifert–Van Kampen theorem of algebraic topology, sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space in terms of the fundamental groups of two open, path-connected subspaces that cover . It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones.

In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane.

In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different categories, as discussed below.

This is a glossary of properties and concepts in category theory in mathematics.

In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes. The concept was introduced by Daniel G. Quillen (1967).

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 coherent equivalence. 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.

In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.

In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid. The 2-groups are part of a larger hierarchy of n-groups. In some of the literature, 2-groups are also called gr-categories or groupal groupoids.

In mathematics, directed algebraic topology is a refinement of algebraic topology for directed spaces, topological spaces and their combinatorial counterparts equipped with some notion of direction. Some common examples of directed spaces are spacetimes and simplicial sets. The basic goal is to find algebraic invariants that classify directed spaces up to directed analogues of homotopy equivalence. For example, homotopy groups and fundamental n-groupoids of spaces generalize to homotopy monoids and fundamental n-categories of directed spaces. Directed algebraic topology, like algebraic topology, is motivated by the need to describe qualitative properties of complex systems in terms of algebraic properties of state spaces, which are often directed by time. Thus directed algebraic topology finds applications in Concurrency, Network traffic control, General Relativity, Noncommutative Geometry, Rewriting Theory, and Biological systems.

Ronald Brown (mathematician)

Ronald Brown is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor University, he has authored many books and more than 160 journal articles.

In mathematics, more specifically category theory, a quasi-category is a generalization of the notion of a category. The study of such generalizations is known as higher category theory.

In category theory, a branch of mathematics, a (left) Bousfield localization of a model category replaces the model structure with another model structure with the same cofibrations but with more weak equivalences.

Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras, simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements. Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for intersection theory of singular algebraic varieties and cotangent complexes in deformation theory, among the other applications.

In category theory, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category of simplicial sets. It is an ∞-category generalization of a groupoid, a category in which every morphism is an isomorphism.

In mathematics, an ∞-topos is, roughly, an ∞-category such that its objects behave like sheaves of spaces with some choice of Grothendieck topology; in other words, it gives an intrinsic notion of sheaves without reference to an external space. The prototypical example of an ∞-topos is the ∞-category of sheaves of spaces on some topological space. But the notion is more flexible; for example, the ∞-category of étale sheaves on some scheme is not the ∞-category of sheaves on any topological space but it is still an ∞-topos.

In mathematics, nonabelian algebraic topology studies an aspect of algebraic topology that involves higher-dimensional algebras.

In Mathematics, an Abelian 2-group is a higher dimensional analogue of an Abelian group, in the sense of higher algebra, which were originally introduced by Alexander Grothendieck while studying abstract structures surrounding Abelian varieties and Picard groups. More concretely, they are given by groupoids which have a bifunctor which acts formally like the addition an Abelian group. Namely, the bifunctor has a notion of commutativity, associativity, and an identity structure. Although this seems like a rather lofty and abstract structure, there are several examples of Abelian 2-groups. In fact, some of which provide prototypes for more complex examples of higher algebraic structures, such as Abelian n-groups.