In Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. One reason is that... mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. Categories appear in virtually every branch of modern mathematics and are a central unifying notion. The study of categories in their own right is known as Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as generalized abstract nonsense. The use of this phrase does not mean that mathematicians consider category theory to be fuzzy or non-rigorous, merely that a... category theory.

For more extensive motivational background and historical notes, see Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as generalized abstract nonsense. The use of this phrase does not mean that mathematicians consider category theory to be fuzzy or non-rigorous, merely that a... category theory and the This is a list of category theory topics, by Wikipedia page. Specific categories Category of sets Concrete category Category of vector spaces Category of graded vector spaces Category of topological spaces Category of metric spaces Category of preordered sets Category of groups Category of abelian groups Category of commutative rings... list of category theory topics.

Definition

A category C consists of

• a In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Some classes are sets, for instance the class of all integers that are even, but others are... class ob(C) of objects:
• a class hom(C) of In mathematics, a morphism is an abstraction of a function or mapping between two spaces. The word can mean different things depending on the type of space in question. In set theory, for example, morphisms are just functions, in group theory they are group homomorphisms, while in topology they are... morphisms. Each morphism f has a unique source object a and target object b. 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)) 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.)

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 every both ob(C) and hom(C) are actually This article is about sets in mathematics. For other meanings, see Set (disambiguation). Sets are one of the most important and fundamental concepts in modern mathematics. Basic set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced... sets. A locally small category is a category such that for all objects a and b, the hom-class hom(a, b) is a set. Many important categories are not small, or even locally small.

The morphisms of a category are sometimes called arrows due to the influence of In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. For example, the first isomorphism theorem is a commutative triangle... commutative diagrams.

Examples

Each category is presented in terms of its objects, its In mathematics, a morphism is an abstraction of a function or mapping between two spaces. The word can mean different things depending on the type of space in question. In set theory, for example, morphisms are just functions, in group theory they are group homomorphisms, while in topology they are... morphisms, and its composition of morphisms.

• The category In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. It is the most basic and the most commonly used category in mathematics. The category is usually denoted simply as Set. The epimorphisms in Set are the surjective maps, the... Set of all This article is about sets in mathematics. For other meanings, see Set (disambiguation). Sets are one of the most important and fundamental concepts in modern mathematics. Basic set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced... sets together with In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). The concept of a function is fundamental to virtually every branch of mathematics... functions between sets, where composition is the usual function composition (The following are A subcategory in Wikipedia is a category that depends on another category. See Wikipedia:Categorization In mathematics, a subcategory of a category C consists of subsets of the morphisms and of the objects of C, such that the subset X of morphisms is closed under composition in C, and the... subcategories of Set, obtained by adding some type of structure onto a set, by requiring that morphisms are functions which respect this added structure, and where morphism composition is simply ordinary function composition.)
  • The category The category Ord has preordered sets as objects and monotonic functions as morphisms. This is a category because the composition of two monotonic functions preserves monotonicity. The monomorphisms in Ord are the injective monotonic functions. The empty set (considered as a preordered set) is the initial object of Ord; any... Ord of all This article is about the mathematics concept. For preorder traversal of a tree data structure, see tree traversal. In mathematics, especially in order theory, preorders are certain kinds of binary relations that are closely related to partially ordered sets. The name quasiorder is also a common expression for preorders. Many... preordered sets with In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. These functions first arose in calculus and were later generalized to the more abstract setting of order theory. Although the concepts generally agree, the two disciplines have developed a slightly different terminology. While in... monotonic functions
  • The category In mathematics, the category of magmas (see category, magma for definitions), denoted by Mag, has as objects sets with a binary operation, and morphisms given by homomorphisms of operations (in the universal algebra sense). The category Mag has direct products, so the concept of a magma object (internal binary operation... Mag consisting of all In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M × M → M. A binary operation is closed by definition, but no other axioms are imposed on the... magmas with their This word should not be confused with homeomorphism. In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure. N.B. Some authors use the word homomorphism in a larger context than that of algebra. Some take... homomorphisms
  • The category In mathematics, the medial category Med, that is, the category of medial magmas has as objects sets with a medial binary operation, and morphisms given by homomorphisms of operations (in the universal algebra sense). The category Med has direct products, so the concept of a medial magma object (internal binary... Med consisting of all For the meaning of medial in anatomy, see anatomical terms of location. In abstract algebra, a medial algebra is a set with a binary operation which satisfies the identity (x . y) . (u . z) = (x . u) . (y . z), or xy.uz=xu.yz Its importance arises in the concept of an... medial magmas with their This word should not be confused with homeomorphism. In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure. N.B. Some authors use the word homomorphism in a larger context than that of algebra. Some take... homomorphisms
  • The category In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. The category Grp is both complete and co-complete. The product is just the direct... Grp consisting of all In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste Galois (1830), concerning the... groups with their Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the... group homomorphisms
  • The category In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category. The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms. The zero... Ab consisting of all In mathematics, an abelian group is a commutative group, i.e. a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. Notation There are two main notational conventions for abelian groups -- additive and multiplicative. Examples Every... abelian groups with their Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the... group homomorphisms
  • The category In mathematics, the category K-Vect has all vector spaces over a fixed field K as objects and linear transformations as morphisms. Since vector spaces over K (as a field) are the same thing as modules over the ring K, K-Vect is a special case of R-Mod, the... Vect_K of all The fundamental concept in linear algebra is that of a vector space or linear space. If one considers geometrical vectors, and the operations one can perform upon these vectors such as addition of vectors, scalar multiplication, with some natural constraints such as closure of these operations, associativity of these and... vector spaces over the In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. Fields are important objects of study in... field K (which is held fixed) with their K- In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... linear maps
  • The category The category Top has topological spaces as objects and continuous maps as morphisms. This is a category because the composition of two continuous maps is again continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. (Some authors... Top of all Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology. Definition A topological space... topological spaces with In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. If small changes in the input can produce a broken jump in the changes of the output (or the value of the ouput is not defined), the function... continuous functions
  • The category The category Met has metric spaces as objects and short maps as morphisms. This is a category because the composition of two short maps is again short. The monomorphisms in Met are the injective short maps, the epimorphisms are the dense image short maps (for instance, the inclusion: , which is... Met of all In mathematics, a metric space is a set (or space) where a distance between points is defined. History Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Circ. Mat. Palermo 22(1906) 1-74. Definition A metric space M is a set of... metric spaces with In mathematics, a short map is a function f from a metric space X to another metric space Y such that for any we have . Here and denote metrics on and , respectively. In other words, f is short iff it is 1-Lipschitz. One can say that f is strictly... short maps
  • The category Uni of all In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. Uniform spaces generalize metric spaces and topological groups and therefore underlie most of analysis. They were introduced by Bourbaki. If X is a set, a nonempty system Φ of subsets of... uniform spaces with In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x effect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but... uniformly continuous functions
• The category Cat of all small categories with functors
• Any This article is about the mathematics concept. For preorder traversal of a tree data structure, see tree traversal. In mathematics, especially in order theory, preorders are certain kinds of binary relations that are closely related to partially ordered sets. The name quasiorder is also a common expression for preorders. Many... 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.)
• Any In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. In other words, it is a unital semigroup. Definition A monoid is a magma (M,*), i.e. a set M with binary operation * : M × M →... monoid 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. In fact, one may view categories as generalizations of monoids; several definitions and theorems about monoids may be generalized for categories.
• Any This article just presents the basic definitions. For a broader view see graph theory. For another mathematical use of graph, see graph of a function. A graph with 6 vertices and 7 edges. In mathematics and computer science a graph is the basic object of study in graph theory. Informally... directed graph generates a small category: the objects are the In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). In graph theory, a graph describes a set of connections between objects. Each object is... vertices of the graph and the morphisms are the paths in the graph. Composition of morphisms is concatenation of paths. This is called the free category generated by the graph.
• If I is a This article is about sets in mathematics. For other meanings, see Set (disambiguation). Sets are one of the most important and fundamental concepts in modern mathematics. Basic set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced... set, the In category theory, a discrete category is a category whose only morphisms are the identity morphisms. It is the simplest kind of category. Specifically a category C is discrete if MorC(X, X) = {idX} for all objects X MorC(X, Y) = ∅ for all objects X ≠ Y Clearly, any... discrete category on I is the small category which has the elements of I as objects and only the identity morphisms as morphisms. 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 In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in... dual or opposite category and is denoted C^op.
• 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 In mathematics, a morphism is an abstraction of a function or mapping between two spaces. The word can mean different things depending on the type of space in question. In set theory, for example, morphisms are just functions, in group theory they are group homomorphisms, while in topology they are... morphism f : a → b is called

• a In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. In the more general (and abstract) setting of category theory, a monomorphism (also called a monic morphism) is a morphism f : X → Y such that f O g1 = f O g2 implies... monomorphism (or monic) if fg₁ = fg₂ implies g₁ = g₂ for all morphisms g₁, g₂ : x → a.
• an In the context of abstract algebra or universal algebra, an epimorphism is simply a homomorphism onto or surjective homomorphism. In the more general (and abstract) setting of category theory, an epimorphism (also called an epic morphism) is a morphism f : X → Y such that g1 O f = g2... epimorphism (or epic) if g₁f = g₂f implies g₁ = g₂ for all morphisms g₁, g₂ : b → x.
• an In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. Douglas Hofstadter provides an informal definition: The word isomorphism applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure... isomorphism if it is both monic and epic; equivalently, if there exists a morphism g : b → a with fg = 1_b and gf = 1_a.
• an In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. So, for example, an endomorphism of a vector space V is a linear map f : V → V and an endomorphism of a group G is a group homomorphism f : G → G... endomorphism if a = b. The class of endomorphisms of a is denoted end(a).
• an 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 mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called... automorphism if f is both an endomorphism and an isomorphism. The class of automorphisms of a is denoted aut(a).

Relations among morphisms (such as fg = h) can most conveniently be represented with In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. For example, the first isomorphism theorem is a commutative triangle... commutative diagrams, where the objects are represented as points and the morphisms as arrows.

Types of categories

• In many categories, the hom-sets hom(a, b) are not just sets but actually In mathematics, an abelian group is a commutative group, i.e. a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. Notation There are two main notational conventions for abelian groups -- additive and multiplicative. Examples Every... abelian groups, and the composition of morphisms is compatible with these group structures; i.e. is In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. For a formal definition, given three vector spaces V, W and X over the same base field F, a bilinear operator is a function B : V × W → X such that for any w in... bilinear. Such a category is called A preadditive category is a category that is enriched over the monoidal category of abelian groups. In other words, the category C is preadditive if every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear over the integers. A... preadditive. If, furthermore, the category has all finite In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces. Essentially, the product of a family of objects is the most general object which admits a morphism to each of... products and In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. Basically, this means the definition is the same as the product but with all arrows reversed. Despite this innocuous-looking change in the name and notation, coproducts can be dramatically different from products. The... coproducts, it is called an In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A1,...,An of C have a biproduct A1 ⊕ ··· ⊕ An in C. (Recall that a category C is preadditive if all its morphism sets are Abelian groups and morphism... additive category. If all morphisms have a 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 kernels from algebra. Intuitively, the kernel of the morphism f : X → Y is the most general morphism k... kernel and a In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f. In a topological setting, one typically takes the closure of the image before passing to the quotient. For instance, if f : H1 → H2 is a... cokernel, and all epimorphism are cokernels and all monomorphisms are kernels, then we speak of an In mathematics, an abelian category is a certain kind of category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. Definitions A category is abelian... abelian category. A typical example of an abelian category is the category of abelian groups.
• A category is called 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 of mathematics, like products and inverse limits. Accordingly, the dual notion of a colimit, generalizes disjoint unions and direct sums. Limits and colimits have... complete if all 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 of mathematics, like products and inverse limits. Accordingly, the dual notion of a colimit, generalizes disjoint unions and direct sums. Limits and colimits have... limits in it exist. The categories of sets, abelian groups and topological spaces are complete.
• A category is called 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. These categories are particularly important in mathematical logic and the theory of programming. For generalizations of this... 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.
• A For discussion of topoi in literary theory, see literary topos. In mathematics, a topos (plural: topoi or toposes - this is a contentious topic) is a type of category which allows the formulation of all of mathematics inside it. Introduction Traditionally, mathematics is built on set theory, and all objects studied... 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). A topos can also be used to represent a logical theory.
• A In mathematics, especially in category theory and homotopy theory, a groupoid is a concept (first developed by Heinrich Brandt) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. They are often used to capture information about geometrical objects such as manifolds. The term groupoid is... groupoid is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups, In mathematics, groups are often used to describe symmetries of objects. This is formalized by the notion of a group action: every element of the group acts like a bijective map (or symmetry) on some set. In this case, the group is also called a transformation group of the set... group actions and In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i.e., if the relation is written as ~ it holds for all a, b and c in X that (Reflexivity) a ~ a (Symmetry) if a ~ b then b ~ a... equivalence relations.

References

• Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990). Abstract and Concrete Categories (http://www.math.uni-bremen.de/~dmb/acc.pdf). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition)
• Asperti, Andrea, & Longo, Giuseppe (1991). Categories, Types and Structures (ftp://ftp.di.ens.fr/pub/users/longo/CategTypesStructures/book.pdf). Originally publ. M.I.T. Press.
• 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.. Vols. 50-52 of Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press.
• Lawvere, William, & Schanuel, Steve. (1997). Conceptual Mathematics: A First Introduction to Categories. Cambridge: Cambridge University Press.
• Mac Lane, Saunders (1998). Categories for the Working Mathematician (2nd ed.). Graduate Texts in Mathematics 5. Springer. ISBN 0-387-98403-8.

See also

• Category theory, Wikibook link
• This is a list of important publications in mathematics, organized by field. Some reasons why a particular publication might be regarded as important: Topic creator – A publication that created a new topic Breakthrough – A publication that changed scientific knowledge significantly Introduction – A publication that is a good... Important publications in category theory

External links

• "Category Theory" in Stanford Encyclopedia of Philosophy (http://plato.stanford.edu/entries/category-theory/)
• Homepage of the Categories mailing list (http://www.mta.ca/~cat-dist/categories.html), with extensive list of resources
• Category Theory section of Alexandre Stefanov's list of free online mathematics resources (http://us.geocities.com/alex_stef/mylist.html)