In mathematics, **categories** allow one to formalize notions involving abstract structure and processes that 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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...
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. ...
This is a list of category theory topics, by Wikipedia page. ...
## Definition
A **category** *C* consists of - a class ob(
*C*) of **objects**: - a class hom(
*C*) of **morphisms**. 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*)) 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: 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. ...
In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...
- (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. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
The morphisms of a category are sometimes called *arrows* due to the influence of commutative diagrams. 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. ...
## 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. (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.) - The category
**Cat** of all small categories with functors. - The category
**Rel** of all sets, with relations as morphisms. - 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.) - Any 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. 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. 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. 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}. - 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. In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ...
In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
In mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions. ...
The category Ord has preordered sets as objects and monotonic functions as morphisms. ...
In mathematics, especially in order theory, preorders are certain kinds of binary relations that are closely related to partially ordered sets. ...
In mathematics, functions between ordered sets are monotonic (or monotone, or even isotone) if they preserve the given order. ...
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). ...
In abstract algebra, a magma (also called a groupoid) is a particularly 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 spaces). ...
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). ...
For the meaning of medial in anatomy, see anatomical terms of location. ...
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ...
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. ...
This picture illustrates how the hours in a clock form a group. ...
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 identity element...
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. ...
In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...
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 identity element...
In mathematics, the category K-Vect has all vector spaces over a fixed field K as objects and linear transformations as morphisms. ...
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 (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
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...
The category Top has topological spaces as objects and continuous maps as morphisms. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...
The category Met has metric spaces as objects and short maps as morphisms. ...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
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 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 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...
In mathematics, the category of manifolds, often denoted Manp, is the category whose objects are manifolds of smoothness class Cp and whose morphisms are p-times continuously differentiable maps. ...
On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
In mathematics, specifically in category theory, the 2-category of small categories is the 2-category whose objects are small categories, whose arrows are functors and whose 2-arrows are natural transformations. ...
Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ...
In mathematics, the category Rel has the class of sets as objects and relations as morphisms. ...
In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
In mathematics, a finitary relation is defined by one of the formal definitions given below. ...
In mathematics, especially in order theory, preorders are certain kinds of binary relations that are closely related to partially ordered sets. ...
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...
This article just presents the basic definitions. ...
This article just presents the basic definitions. ...
In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
In category theory, a discrete category is a category whose only morphisms are the identity morphisms. ...
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...
## Types of morphisms A morphism *f* : *a* → *b* is called In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...
- a
*monomorphism* (or *monic*) if *fg*_{1} = *fg*_{2} implies *g*_{1} = *g*_{2} for all morphisms *g*_{1}, *g*_{2} : *x* → *a*. - an
*epimorphism* (or *epic*) if *g*_{1}f = *g*_{2}f implies *g*_{1} = *g*_{2} for all morphisms *g*_{1}, *g*_{2} : *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* if it has a left inverse, i.e. if there exists a morphism *g* : *b* → *a* with *gf* = 1_{a}. - an
*isomorphism* 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* if *a* = *b*. The class of endomorphisms of *a* is denoted end(*a*). - an
*automorphism* 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: In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ...
In category theory an epimorphism (also called an epic morphism or an epi) is a morphism f : X â†’ Y which is right-cancellable in the following sense: g1 o f = g2 o f implies g1 = g2 for all morphisms g1, g2 : Y â†’ Z. Epimorphisms are analogues of surjective functions, but...
A retraction is a public statement that confirms that a previously made statement was incorrect, invalid, or morally wrong. ...
In the mathematical field of category theory, a section is a morphism which has a left inverse, i. ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...
In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ...
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...
*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, 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. ...
## 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.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 epimorphism 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. - A category is called complete if all limits exist in it. 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.
- 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). A topos can also be used to represent a logical theory.
- A groupoid is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups, group actions and equivalence relations.
In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...
In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...
A preadditive category is a category that is enriched over the monoidal category of abelian groups. ...
In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ...
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...
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. ...
In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f. ...
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ...
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. ...
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. ...
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. ...
In mathematics, a topos (plural topoi or toposes) is a type of category that behaves like the category of sheaves of sets on a topological space. ...
In mathematics, especially in category theory and homotopy theory, a groupoid is a concept (first developed by Heinrich Brandt in 1926) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. ...
In mathematics, a symmetry group describes all symmetries of objects. ...
In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ...
## References - Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990).
*Abstract and Concrete Categories*. Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition) - Asperti, Andrea, & Longo, Giuseppe (1991).
*Categories, Types and Structures*. Originally publ. M.I.T. Press. - Barr, Michael, & Wells, Charles (2002).
*Toposes, Triples and Theories*. (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. ## External links |