Category (mathematics)

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Category (mathematics)

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a category is a fundamental and abstract way to describe mathematical entities and their relationships. A category is composed of a collection of abstract "objects" of any kind, linked together by a collection of abstract "arrows

Morphism

In mathematics, a morphism is an Abstraction derived from structure-preserving map between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory....
" of any kind that have a few basic properties (the ability to compose the arrows associatively

Associativity

In mathematics, associativity is a property that a binary operation can have. It means that, within an expression containing two or more of the same associative operators in a row, the order that the operations are performed does not matter as long as the sequence of the operands is not changed....
and the existence of an identity arrow for each object). 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 two arrows.

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In mathematics

Mathematics

Morphism

Associativity

Equivalence of categories

In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same"....
" for purposes of category theory, even if they are not precisely the same. Many well-known categories are conventionally identified by a short capitalized word or abbreviation in bold or italics such as Set (category of sets

Category of sets

In mathematics, the category of sets, denoted as Set, is the Category theory whose Category theory are all Set and whose morphisms are all function s....
) or Ring (category of rings

Category of rings

In mathematics, the category of rings, denoted by Ring, is the category whose objects are ring and whose morphisms are ring homomorphisms ....
).

The notion of a category is the central idea within a branch of mathematics called category theory
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
, which seeks to generalize all of mathematics in terms of such abstract objects and arrows, independent of the particular details of what the 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. For more extensive motivational background and historical notes, see category theory

Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
and the list of category theory topics

List of category theory topics

This is a list of category theory topics, by Wikipedia page....
.

Definition

A category C consists of

a class
Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of Set which can be unambiguously defined by a property that all its members share....
ob(C) of objects:
a class hom(C) of morphism
Morphism

In mathematics, a morphism is an Abstraction derived from structure-preserving map between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory....
s, or arrow
Commutative diagram

In mathematics, and especially in category theory a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition....
s, or map
Map (mathematics)

In mathematics and related technical fields, the term map or mapping is often a synonym for Function . Thus, for example, a partial map is a partial function, and a total map is a total function....
s, between the objects. Each morphism f has a unique source object a and target object b where a and b are in ob(C). 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 which category hom(a, b) refers to) to denote the hom-class of all morphisms from a to b. (Some authors write Mor(a, b) or simply C(a, b) instead.)
for every three objects a, b and c, a binary operation hom(a, b) × hom(b, c) ? hom(a, c) called composition of morphisms; the composition of f : a ? b and g : b ? c is written as g o f or gf. (Some authors write fg or f;g.)

such that the following axioms hold:

(associativity) if f : a ? b, g : b ? c and h : c ? d then h o (g o f) = (h o g) o f, and
(identity) for every object x, there exists a morphism 1_x : x ? x called the identity morphism for x, such that for every morphism f : a ? b, we have 1_b o f = f = f o 1_a.

From these axioms, one can prove that there is exactly one identity morphism for every object. Some authors use a slight variation of the definition in which each object is identified with the corresponding identity morphism.

A small category is a category in which both ob(C) and hom(C) are actually sets and not proper classes. A category that is not small is said to be large. 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.

The morphisms of a category are sometimes called arrows due to the influence of commutative diagram

Commutative diagram

In mathematics, and especially in category theory a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition....
s.

Examples

The category Set

Category of sets

In mathematics, the category of sets, denoted as Set, is the Category theory whose Category theory are all Set and whose morphisms are all function s....
consists of all sets together with all functions

Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
between sets, where composition is the usual function composition

Function composition

In mathematics, a composite function represents the application of one function to the results of another. For instance, the functions and can be composed by first computing a f and then applying a function g to the output of f....
. Set is a large category.

Any directed graph

Directed graph

A directed graph or digraph is a pair G= of:* a Set V, whose element are called vertices or nodes,* a set A of ordered pairs of vertices, called arcs, directed edges, or arrows....
generates

Generating set

In mathematics, the expressions generator, generate, generated by and generating set can have several closely related technical meanings:...
a small category: the objects are the vertices

Vertex (graph theory)

In graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs ....
of the graph, and the morphisms are the paths in the graph where composition of morphisms is concatenation of paths. Such a category is called the free
Free object

In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure ....
category generated by the graph. This example demonstrates that morphisms need not be functions.

A discrete category
Discrete category

In mathematics, especially category theory, a discrete category is a category whose only morphisms are the identity morphisms. It is the simplest kind of category....
is a category whose only morphisms are the identity morphisms. If I is a set, the discrete category on I is the small category that has the elements of I as objects and only the identity morphisms as morphisms. The composition law is forced, because there is at most one morphism from any object to another. Discrete categories are the simplest kind of category.

Any monoid

Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element....
forms a small category with a single object x. (Here, x is any fixed set.) The morphisms from x to x are precisely the elements of the monoid, and the categorical composition of morphisms is given by the monoid operation. One may view categories as generalizations of monoids; several definitions and theorems about monoids may be generalized for categories.

Any group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
can be seen as a category with a single object in which every morphism is invertible (for every morphism f there is a morphism g that is both left and right inverse

Morphism

In mathematics, a morphism is an Abstraction derived from structure-preserving map between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory....
to f under composition) by viewing the group as acting

Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
on itself by left multiplication.

Any preordered set

Preorder

In mathematics, especially in order theory, preorders are binary relations that satisfy certain conditions. For example, all partial orders and equivalence relations are preorders....
(P, =) forms a small category, where the objects are the members of P, the morphisms are arrows pointing from x to y when x = y. Between any two objects there can be at most one morphism. The existence of identity morphisms and the composability of the morphisms are guaranteed by the reflexivity

Reflexive relation

In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity.At least in this context, relation always means a subset of X ? X....
and the transitivity

Transitive relation

In mathematics, a binary relation R over a Set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
of the preorder. By the same argument, any partially ordered set

Partially ordered set

In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set ....
and any equivalence relation

Equivalence relation

In mathematics, an equivalence relation is, loosely, a binary relation on a Set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets....
can be seen as a small category. Any ordinal number

Ordinal number

In set theory, an ordinal number, or just ordinal, is the order type of a well-order. They are usually identified with hereditarily transitive sets....
can be seen as a category when viewed as a ordered set

Total order

In mathematics and set theory, a total order, linear order, simple order, or ordering is a binary relation on some Set X....
.

The category Rel

Category of relations

In mathematics, the category Rel has the class of Set as object and binary relations as morphism .A morphism R : A ? B in this category is a relation between the sets A and B, so ....
consists of all sets, with binary relation

Binary relation

In mathematics, a binary relation is an arbitrary association of elements within a set or with elements of another set.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that is a divisibility of p, and no othe...
s as morphisms. Abstracting from relations

Relation (mathematics)

In mathematics , a relation is a property that assigns truth values to combinations of k first-order logic. Typically, the property describes a possible connection between the components of a k-tuple....
instead of functions yields allegories

Allegory (category theory)

In mathematics, in the subject of category theory, an allegory is a category that has some of the structure of the category of sets and binary relations between them....
instead of categories.

The category Cat

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 category and whose morphisms are functors between categories....
consists of all small categories with 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 morphisms in the category of small categories....
s.

Concrete categories

The following are examples of concrete categories

Concrete category

In mathematics, a concrete category is commonly understood as a category whose objects are mathematical structure Set , whose morphisms are structure-preserving function s, and whose composition operation is function composition....
, obtained by adding some type of structure onto Set, and requiring that morphisms are functions that respect this added structure; the morphism composition is ordinary function composition.

Category	Objects	Morphisms
Ord Category of preordered sets The category theory Ord has Preorder as object and monotonic functions as morphisms. This is a category because the function composition of two monotonic functions is monotonic and the identity map is monotonic....	preordered sets Preorder In mathematics, especially in order theory, preorders are binary relations that satisfy certain conditions. For example, all partial orders and equivalence relations are preorders....	monotonic function Monotonic function In mathematics, a monotonic function is a function which preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.... s
Mag	magma Magma (algebra) In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a Set M equipped with a single binary operation M × M ? M.... s	magma homomorphisms
Med	medial magmas	magma homomorphisms
Grp Category of groups In mathematics, the category theory Grp has the class of all Group for objects and group homomorphisms for morphisms. As such, it is a concrete category....	group Group (mathematics) In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element.... s	group homomorphism Group homomorphism In mathematics, given two group and , a group homomorphism from to is a function h : G ? H such that for all u and v in G it holds that... s
Ab Category of abelian groups In mathematics, the category theory Ab has the abelian groups as object and group homomorphisms as morphisms. This is the prototype of an abelian category....	abelian group Abelian group An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order .... s	group homomorphisms
Ring Category of rings In mathematics, the category of rings, denoted by Ring, is the category whose objects are ring and whose morphisms are ring homomorphisms ....	ring Ring (mathematics) In mathematics, a ring is a type of algebraic structure. There is some variation among mathematicians as to exactly what properties a ring is required to have, as described in detail below.... s	ring homomorphism Ring homomorphism In ring theory or abstract algebra, a ring homomorphism is a function between two ring which respects the operations of addition and multiplication.... s
R-Mod	R-Modules, where R is a Ring	module homomorphisms
Vect_K	vector space Vector space File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context.... s over the field Field (mathematics) In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms.... K	K-linear maps
Top Category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose object s are topological spaces and whose morphisms are continuous maps....	topological space Topological space Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function .... s	continuous function Continuous function (topology) In topology and related areas of mathematics a continuous function is a morphism between topological spaces. Intuitively, this is a function f where a set of points near f always contain the of a set of points near x.... s
Met Category of metric spaces The category theory Met, first considered by Isbell , has metric spaces as object and metric maps or short maps as morphisms. This is a category because the function composition of two metric maps is again metric....	metric space Metric space In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space.... s	short map Short map In the mathematics theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance .These maps are the morphisms in the category of metric spaces, Met .... s
Uni	uniform space Uniform space In the mathematical field of topology, a uniform space is a Set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform property such as complete space, uniform continuity and uniform convergence.... s	uniformly continuous functions
Man^p Category of manifolds In mathematics, the category of manifolds, often denoted Manp, is the category whose object s are manifolds of smooth function Cp and whose morphisms are p-times continuously differentiable maps....	smooth manifolds	p-times continuously differentiable maps

Fiber bundle

File:Roundhairbrush.JPGIn mathematics, and particularly topology, a fiber bundle is intuitively a space E which locally "looks" like a product space B ? F, but globally may have a different topological structure....
s with bundle map

Bundle map

In mathematics, a bundle map is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common fiber bundle....
s between them form a concrete category.

Construction of new categories

Dual category

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

Dual (category theory)

In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop....
and is denoted C^op.

Product categories

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.

Types of morphisms

A morphism

Morphism

a monomorphism
Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an Injective function homomorphism. A monomorphism from X to Y is often denoted with the notation ....
(or monic) if fg₁ = fg₂ implies g₁ = g₂ for all morphisms g₁, g₂ : x ? a.
an epimorphism
Epimorphism

In category theory an epimorphism is a morphism f : X ? Y which is Cancellation property in the sense that, for all morphisms ,Epimorphisms are analogues of surjective functions, but they are not exactly the same....
(or epic) if g₁f = g₂f implies g₁ = g₂ for all morphisms g₁, g₂ : b ? x.
a bimorphism if it is both a monomorphism and an epimorphism.
a retraction if it has a right inverse, i.e. if there exists a morphism g : b ? a with fg = 1_b.
a section
Section (category theory)

In category theory, a branch of mathematics, a section is a right inverse of a morphism. Dually, a retraction is a left inverse. In other words, if and are morphisms whose composition is the identity function on Y, then g is a section of f, and f is a retraction of g....
if it has a left inverse, i.e. if there exists a morphism g : b ? a with gf = 1_a.
an isomorphism
Isomorphism

In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings....
if it has an inverse, i.e. if there exists a morphism g : b ? a with fg = 1_b and gf = 1_a.
an endomorphism
Endomorphism

In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ?: V ? V, and an endomorphism of a group G is a group homomorphism ?: G ? G....
if a = b. The class of endomorphisms of a is denoted end(a).
an automorphism
Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of map the object to itself while preserving all of its structure....
if f is both an endomorphism and an isomorphism. The class of automorphisms of a is denoted aut(a).

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

f is a monomorphism and a retraction;
f is an epimorphism and a section;
f is an isomorphism.

Relations among morphisms (such as fg = h) can most conveniently be represented with commutative diagram

Commutative diagram

Types of categories

In many categories, the hom-sets hom(a, b) are not just sets but actually abelian group
Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
s, and the composition of morphisms is compatible with these group structures; i.e. is bilinear
Bilinear

Bilinear may refer to* Bilinear sampling, a method in computer graphics for choosing the color of a texture.* Bilinear form* Bilinear interpolation...
. Such a category is called preadditive
Preadditive category

In mathematics, specifically in category theory, a preadditive category is a category that is enriched category over the monoidal category of abelian groups....
. If, furthermore, the category has all 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, the direct product of groups, the direct product of rings and the product topology....
and coproduct
Coproduct

In category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union and disjoint union , the free product, and the direct sum of modules and vector spaces....
s, it is called an additive category
Additive category

In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A1,...,A'n of C have a biproduct A1 ? ? ? A'n in C....
. If all morphisms have a kernel
Kernel (category theory)

In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernel ....
and a cokernel
Cokernel

In mathematics, the cokernel of a linear mapping of vector spaces f : X ? Y is the quotient space Y/im of the codomain of f by the image of f....
, and all epimorphisms are cokernels and all monomorphisms are kernels, then we speak of an abelian category
Abelian category

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernel s and cokernels exist and have desirable properties....
. A typical example of an abelian category is the category of abelian groups.
A category is called complete
Complete category

In mathematics, a complete category is a category in which all small limit s exist. That is, a category C is complete if every diagram F : J → C where J is small category has a limit in C....
if all limits
Limit (category theory)

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as product and inverse limits....
exist in it. The categories of sets, abelian groups and topological spaces are complete.
A category is called cartesian closed
Cartesian closed category

In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors....
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.
A topos
Topos

In mathematics, a topos is a type of category that behaves like the category of sheaf theory of Set on a topological space. For a discussion of the history of topos theory, see the article Background and genesis of topos theory....
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.
A groupoid
Groupoid

In abstract algebra, a branch of mathematics, especially in category theory and homotopy theory, a 'groupoid' generalises the notion of group and of category in several equivalent ways....
is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups, group action
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
s and equivalence relation
Equivalence relation

In mathematics, an equivalence relation is, loosely, a binary relation on a Set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets....
s.

External links

Chris Hillman, , formal introduction to Category Theory.
, with extensive list of resources

Category (mathematics)

Definition

Examples

Concrete categories

Construction of new categories

Dual category

Product categories

Types of morphisms

Types of categories

See also

External links