In mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A category is composed of a collection of abstract "objects" of any kind, linked together by a collection of abstract "arrows" of any kind that have a few basic properties (the ability to compose the arrows associatively and the existence of an identity arrow for each object). In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, associativity is a property that a Binary operation can have 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. Two categories may also be considered "equivalent" for purposes of category theory, even if they are not precisely the same. In Category theory, an abstract branch of Mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are Many well-known categories are conventionally identified by a short capitalized word or abbreviation in bold or italics such as Set (category of sets) or Ring (category of rings). In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are In Mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity and whose Morphisms

The notion of a category is the central idea within a branch of mathematics called category theory, 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. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets 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 and the list of category theory topics. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets This is a list of Category theory topics, by Wikipedia page Specific categories Category of sets Concrete category

Contents 1 Definition 2 Examples 3 Types of morphisms 4 Types of categories 5 See also 6 References 7 External links

Definition

A category C consists of

• a class ob(C) of objects:
• a class hom(C) of morphisms, or arrow between the objects. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also 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 (or arrow) from a to b". We write hom(a, b) (or hom_C(a, b)) to denote the hom-class of all morphisms from a to b. (Some authors write Mor(a, b). )
• 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 diagrams. In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also

Examples

Each category is presented in terms of its objects, its morphisms, and its composition of morphisms.

• The category Set of all sets together with functions between sets, where composition is the usual function composition. In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function (The following are examples of concrete categories, obtained by adding some type of structure onto Set, and requiring that morphisms are functions that respect this added structure; the morphism composition is simply ordinary function composition. In Mathematics, a concrete category is commonly understood as a category whose objects are structured sets, whose Morphisms are structure-preserving )
  • The category Ord of all preordered sets with monotonic functions
  • The category Mag consisting of all magmas with their homomorphisms
  • The category Med consisting of all medial magmas with their homomorphisms
  • The category Grp consisting of all groups with their group homomorphisms
  • The category Ab consisting of all abelian groups with their group homomorphisms
  • The category Ring consisting of all rings with their ring homomorphisms
  • The category Vect_K of all vector spaces over the field K (which is held fixed) with their K-linear maps
  • The category Top of all topological spaces with continuous functions
  • The category Met of all metric spaces with short maps
  • The category Uni of all uniform spaces with uniformly continuous functions
  • The category Man^p of all smooth manifolds with p-times continuously differentiable maps. The category Ord has preordered sets as objects and Monotonic functions as Morphisms This is a category because the composition In Mathematics, especially in Order theory, preorders are Binary relations that satisfy certain conditions In Mathematics, the category of magmas (see category, magma for definitions denoted by Mag, has as objects sets with a Binary operation In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In Mathematics, the medial category Med, that is the category of Medial magmas has as objects sets with a Medial Binary This article is about medial in mathematics For other uses see Medial (disambiguation. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the category Ab has the Abelian groups as objects and Group homomorphisms as Morphisms This is the prototype An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity and whose Morphisms In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication In Mathematics, especially Category theory, the category K-Vect has all Vector spaces over a fixed field K as objects In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, the category of topological spaces, often denoted Top, is the category whose objects are Topological spaces and whose Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function The category Met, first considered by Isbell (1964 has Metric spaces as objects and Metric maps or Short maps as Morphisms In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In the mathematical theory of Metric spaces a metric map or short map is a Continuous function between metric spaces that does not increase any In the Mathematical field of Topology, a uniform space is a set with a uniform structure. In the Mathematical field of Topology, a uniform space is a set with a uniform structure. In Mathematical analysis, a function f ( x) is called uniformly continuous if roughly speaking small changes in the input x effect In Mathematics, the category of manifolds, often denoted Man p, is the category whose objects are Manifolds of A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be
• The category Cat of all small categories with functors. In Mathematics, specifically in Category theory, the category of small categories, denoted by Cat, is the category whose objects are all In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories
• The category Rel of all sets, with relations as morphisms. In Mathematics, the category Rel has the class of sets as objects and Binary relations as morphisms. This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations
• Any preordered set (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 (The composition law is forced, because there is at most one morphism from any object to another. In Mathematics, especially in Order theory, preorders are Binary relations that satisfy certain conditions )
• Any monoid forms a small category with a single object x. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation (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. The monoid demonstrates that morphisms need not be functions, as here, the only function from the singleton set x to x is a trivial function. One may view categories as generalizations of monoids; several definitions and theorems about monoids may be generalized for categories.
• Any directed graph generates a small category: the objects are the vertices of the graph and the morphisms are the paths in the graph. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. For other uses see Vertex. In Graph theory, a vertex (plural vertices) or node is the fundamental unit out Composition of morphisms is concatenation of paths. This is called the free category generated by the graph.
• 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. In Mathematics, especially Category theory, a discrete category is a category whose only Morphisms are the Identity morphisms It is the simplest Again, the composition law is forced. )
• 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. In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the
• 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 f : a → b is called

• a monomorphism (or monic) if fg₁ = fg₂ implies g₁ = g₂ for all morphisms g₁, g₂ : x → a. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism.
• an epimorphism (or epic) if g₁f = g₂f implies g₁ = g₂ for all morphisms g₁, g₂ : b → x. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which
• a bimorphism if it is both a monomorphism and an epimorphism.
• a retraction if it has a right inverse, i. In Category theory, a branch of Mathematics, a section is a right inverse of a morphism e. if there exists a morphism g : b → a with fg = 1_b.
• a section if it has a left inverse, i. In Category theory, a branch of Mathematics, a section is a right inverse of a morphism e. if there exists a morphism g : b → a with gf = 1_a.
• an isomorphism if it has an inverse, i. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective e. if there exists a morphism g : b → a with fg = 1_b and gf = 1_a.
• an endomorphism if a = b. In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself The class of endomorphisms of a is denoted end(a).
• an automorphism if f is both an endomorphism and an isomorphism. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself 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 diagrams, where the objects are represented as points and the morphisms as arrows. In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also

Types of categories

• In many categories, 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. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the e. is bilinear. Such a category is called preadditive. In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category If, furthermore, the category has all finite products and coproducts, it is called an additive category. In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological In Mathematics, specifically in Category theory, an additive category is a Preadditive category C such that any finitely many objects A 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. In Category theory and its applications to other branches of Mathematics, kernels are a generalization of the kernels of Group homomorphisms and the kernels In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist A typical example of an abelian category is the category of abelian groups.
• A category is called complete if all limits exist in it. In Mathematics, a complete category is a category in which all small limits exist In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts The categories of sets, abelian groups and topological spaces are complete.
• A category is called cartesian closed 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. 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
• A topos 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). In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets A topos can also be used to represent a logical theory.
• A groupoid is a category in which every morphism is an isomorphism. In Mathematics, especially in Category theory and Homotopy theory Groupoids are generalizations of groups, group actions and equivalence relations. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent"

See also

• Enriched category
• Higher category theory

References

• Adámek, Jiří; Herrlich, Horst & Strecker, George E. In Category theory and its applications to Mathematics, an enriched category is a category whose Hom-sets are replaced by objects from some other 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 (1990), Abstract and Concrete Categories, John Wiley & Sons, ISBN 0-471-60922-6, <http://katmat.math.uni-bremen.de/acc/acc.pdf>  (now free on-line edition, GNU FDL). The GNU Free Documentation License ( GNU FDL or simply GFDL) is a Copyleft License for free documentation designed by the Free Software
• Asperti, Andrea & Longo, Giuseppe (1991), Categories, Types and Structures, MIT Press, <ftp://ftp.di.ens.fr/pub/users/longo/CategTypesStructures/book.pdf> .
• Barr, Michael & Wells, Charles (2002), Toposes, Triples and Theories, <http://www.cwru.edu/artsci/math/wells/pub/ttt.html>  (revised and corrected free online version of Grundlehren der mathematischen Wissenschaften (278) Springer-Verlag, 1983).
• Borceux, Francis (1994), “Handbook of Categorical Algebra”, Encyclopedia of Mathematics and its Applications, vol. 50–52, Cambridge: Cambridge University Press .
• Lawvere, William & Schanuel, Steve (1997), Conceptual Mathematics: A First Introduction to Categories, Cambridge: Cambridge University Press . Francis William Lawvere (b February 9 1937 in Muncie Indiana is a Mathematician known for his work in Category theory, topos theory and the Philosophy
• Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed. Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American ), Graduate Texts in Mathematics 5, Springer-Verlag, ISBN 0-387-98403-8 .
• Marquis, Jean-Pierre (2006), “Category Theory”, in Zalta, Edward N. , Stanford Encyclopedia of Philosophy, <http://plato.stanford.edu/entries/category-theory/> .

External links

• Homepage of the Categories mailing list, with extensive list of resources
• Category Theory section of Alexandre Stefanov's list of free online mathematics resources

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