Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions.

For other uses, see Category (disambiguation) § Mathematics.

Category theory is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships.

In addition to formalizing mathematics, category theory is also used to formalize many other systems in computer science, such as the semantics of programming languages.

Two categories are the same if they have the same collection of objects, the same collection of arrows, and the same associative method of composing any pair of arrows. Two different categories may also be considered "equivalent" for purposes of category theory, even if they do not have precisely the same structure.

Well-known categories are denoted by a short capitalized word or abbreviation in bold or italics: examples include Set, the category of sets and set functions; Ring, the category of rings and ring homomorphisms; and Top, the category of topological spaces and continuous maps. All of the preceding categories have the identity map as identity arrows and composition as the associative operation on arrows.

The classic and still much used text on category theory is Categories for the Working Mathematician by Saunders Mac Lane. Other references are given in the References below. The basic definitions in this article are contained within the first few chapters of any of these books.

Any monoid can be understood as a special sort of category (with a single object whose self-morphisms are represented by the elements of the monoid), and so can any preorder.

a ob(C) of objects,

class

a class mor(C) of or arrows,

morphisms

a domain or source class function dom: mor(C) → ob(C),

a codomain or target class function cod: mor(C) → ob(C),

for every three objects a, b and c, a binary operation hom(a, b) × hom(b, c) → hom(a, c) called composition of morphisms. Here hom(a, b) denotes the subclass of morphisms f in mor(C) such that dom(f) = a and cod(f) = b. Morphisms in this subclass are written f : a → b, and the composite of f : a → b and g : b → c is often written as g ∘ f or gf.

There are many equivalent definitions of a category.^[1] One commonly used definition is as follows. A category C consists of

such that the following axioms hold:

We write f: a → b, and we say "f is a morphism from a to b". We write hom(a, b) (or hom_C(a, b) when there may be confusion about to which category hom(a, b) refers) to denote the hom-class of all morphisms from a to b.^[2]

Some authors write the composite of morphisms in "diagrammatic order", writing f;g or fg instead of g ∘ f.

From these axioms, one can prove that there is exactly one identity morphism for every object. Often the map assigning each object its identity morphism is treated as an extra part of the structure of a category, namely a class function i: ob(C) → mor(C). Some authors use a slight variant of the definition in which each object is identified with the corresponding identity morphism. This stems from the idea that the fundamental data of categories are morphisms and not objects. In fact, categories can be defined without reference to objects at all using a partial binary operation with additional properties.

Small and large categories[edit]

A category C is called small if both obj(C) and hom(C) are actually sets and not proper classes, and large otherwise. A locally small category is a category such that for all objects a and b, the hom-class hom(a, b) is a set, called a homset. Many important categories in mathematics (such as the category of sets), although not small, are at least locally small. Since, in small categories, the objects form a set, a small category can be viewed as an algebraic structure similar to a monoid but without requiring closure properties. Large categories on the other hand can be used to create "structures" of algebraic structures.

Construction of new categories[edit]

Dual category[edit]

Any category C can itself be considered as a new category in a different way: the objects are the same as those in the original category but the arrows are those of the original category reversed. This is called the dual or opposite category and is denoted C^op.

Product categories[edit]

If C and D are categories, one can form the product category C × D: the objects are pairs consisting of one object from C and one from D, and the morphisms are also pairs, consisting of one morphism in C and one in D. Such pairs can be composed componentwise.

a (or monic) if it is left-cancellable, i.e. fg₁ = fg₂ implies g₁ = g₂ for all morphisms g₁, g₂ : x → a.

monomorphism

an (or epic) if it is right-cancellable, i.e. g₁f = g₂f implies g₁ = g₂ for all morphisms g₁, g₂ : b → x.

epimorphism

a if it is both a monomorphism and an epimorphism.

bimorphism

a if it has a right inverse, i.e. if there exists a morphism g : b → a with fg = 1_b.

retraction

a if it has a left inverse, i.e. if there exists a morphism g : b → a with gf = 1_a.

section

an if it has an inverse, i.e. if there exists a morphism g : b → a with fg = 1_b and gf = 1_a.

isomorphism

an if a = b. The class of endomorphisms of a is denoted end(a). For locally small categories, end(a) is a set and forms a monoid under morphism composition.

endomorphism

an if f is both an endomorphism and an isomorphism. The class of automorphisms of a is denoted aut(a). For locally small categories, it forms a group under morphism composition called the automorphism group of a.

automorphism

A morphism f : a → b is called

Every retraction is an epimorphism. Every section is a monomorphism. The following three statements are equivalent:

Relations among morphisms (such as fg = h) can most conveniently be represented with commutative diagrams, where the objects are represented as points and the morphisms as arrows.

In many categories, e.g. or Vect_K, the hom-sets hom(a, b) are not just sets but actually abelian groups, and the composition of morphisms is compatible with these group structures; i.e. is bilinear. Such a category is called preadditive. If, furthermore, the category has all finite products and coproducts, it is called an additive category. If all morphisms have a kernel and a cokernel, and all epimorphisms are cokernels and all monomorphisms are kernels, then we speak of an abelian category. A typical example of an abelian category is the category of abelian groups.

Ab

A category is called if all small limits exist in it. The categories of sets, abelian groups and topological spaces are complete.

complete

A category is called if it has finite direct products and a morphism defined on a finite product can always be represented by a morphism defined on just one of the factors. Examples include Set and CPO, the category of complete partial orders with Scott-continuous functions.

cartesian closed

A is a certain type of cartesian closed category in which all of mathematics can be formulated (just like classically all of mathematics is formulated in the category of sets). A topos can also be used to represent a logical theory.