# Category (mathematics)

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

## Contents

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

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

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

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

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

Group-like structures
Totality α Associativity Identity Invertibility Commutativity
Semigroupoid UnneededRequiredUnneededUnneededUnneeded
Small Category UnneededRequiredRequiredUnneededUnneeded
Groupoid UnneededRequiredRequiredRequiredUnneeded
Magma RequiredUnneededUnneededUnneededUnneeded
Quasigroup RequiredUnneededUnneededRequiredUnneeded
Unital Magma RequiredUnneededRequiredUnneededUnneeded
Loop RequiredUnneededRequiredRequiredUnneeded
Semigroup RequiredRequiredUnneededUnneededUnneeded
Inverse Semigroup RequiredRequiredUnneededRequiredUnneeded
Monoid RequiredRequiredRequiredUnneededUnneeded
Commutative monoid RequiredRequiredRequiredUnneededRequired
Group RequiredRequiredRequiredRequiredUnneeded
Abelian group RequiredRequiredRequiredRequiredRequired
Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently.

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

## Definition

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

• a class ob(C) of objects,
• a class hom(C) of morphisms , or arrows, or maps between the objects,
• a domain, or source object class function ${\displaystyle \mathrm {dom} \colon \mathrm {hom} (C)\rightarrow \mathrm {ob} (C)}$,
• a codomain, or target object class function ${\displaystyle \mathrm {cod} \colon \mathrm {hom} (C)\rightarrow \mathrm {ob} (C)}$,
• for every three objects a, b and c, a binary operation hom(a, b) × hom(b, c) → hom(a, c) called composition of morphisms; the composition of f : ab and g : bc is written as gf or gf. (Some authors use "diagrammatic order", writing f;g or fg).

Note: Here hom(a, b) denotes the subclass of morphisms f in hom(C) such that ${\displaystyle \mathrm {dom} (f)=a}$ and ${\displaystyle \mathrm {cod} (f)=b}$. Such morphisms are often written as f : ab.

such that the following axioms hold:

• (associativity) if f : ab, g : bc and h : cd then h ∘ (gf) = (hg) ∘ f, and
• (identity) for every object x, there exists a morphism 1x : xx (some authors write idx) called the identity morphism for x, such that every morphism f : ax satisfies 1xf = f, and every morphism g : xb satisfies g ∘ 1x = g.

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

## Small and large categories

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

## Examples

The class of all sets (as objects) together with all functions between them (as morphisms), where the composition of morphisms is the usual function composition, forms a large category, Set . It is the most basic and the most commonly used category in mathematics. The category Rel consists of all sets (as objects) with binary relations between them (as morphisms). Abstracting from relations instead of functions yields allegories, a special class of categories.

Any class can be viewed as a category whose only morphisms are the identity morphisms. Such categories are called discrete. For any given set I, the discrete category on I is the small category that has the elements of I as objects and only the identity morphisms as morphisms. Discrete categories are the simplest kind of category.

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 xy. Furthermore, if is antisymmetric, there can be at most one morphism between any two objects. The existence of identity morphisms and the composability of the morphisms are guaranteed by the reflexivity and the transitivity of the preorder. By the same argument, any partially ordered set and any equivalence relation can be seen as a small category. Any ordinal number can be seen as a category when viewed as an ordered set.

Any monoid (any 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, the identity morphism of x is the identity of the monoid, and the categorical composition of morphisms is given by the monoid operation. Several definitions and theorems about monoids may be generalized for categories.

Similarly any group can be seen as a category with a single object in which every morphism is invertible, that is, for every morphism f there is a morphism g that is both left and right inverse to f under composition. A morphism that is invertible in this sense is called an isomorphism.

A groupoid is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups, group actions and equivalence relations. Actually, in the view of category the only difference between groupoid and group is that a groupoid may have more than one object but the group must have only one. Consider a topological space X and fix a base point ${\displaystyle x_{0}}$ of X, then ${\displaystyle \pi _{1}(X,x_{0})}$ is the fundamental group of the topological space X and the base point ${\displaystyle x_{0}}$, and as a set it has the structure of group; if then let the base point ${\displaystyle x_{0}}$ runs over all points of X, and take the union of all ${\displaystyle \pi _{1}(X,x_{0})}$, then the set we get has only the structure of groupoid (which is called as the fundamental groupoid of X): two loops (under equivalence relation of homotopy) may not have the same base point so they can not multiple with each other. In the language of category, this means here two morphisms may not have the same source object (or target object, because in this case for any morphism the source object and the target object are same: the base point) so they can not compose with each other.

Any directed graph generates a small category: the objects are the vertices of the graph, and the morphisms are the paths in the graph (augmented with loops as needed) where composition of morphisms is concatenation of paths. Such a category is called the free category generated by the graph.

The class of all preordered sets with monotonic functions as morphisms forms a category, Ord . It is a concrete category, i.e. a category obtained by adding some type of structure onto Set, and requiring that morphisms are functions that respect this added structure.

The class of all groups with group homomorphisms as morphisms and function composition as the composition operation forms a large category, Grp . Like Ord, Grp is a concrete category. The category Ab , consisting of all abelian groups and their group homomorphisms, is a full subcategory of Grp, and the prototype of an abelian category. Other examples of concrete categories are given by the following table.

CategoryObjectsMorphisms
Grp groups group homomorphisms
Mag magmas magma homomorphisms
Manp smooth manifolds p-times continuously differentiable maps
Met metric spaces short maps
R-Mod R-modules, where R is a ring R-module homomorphisms
Mon monoids monoid homomorphisms
Ring rings ring homomorphisms
Set sets functions
Top topological spaces continuous functions
Uni uniform spaces uniformly continuous functions
VectK vector spaces over the field KK-linear maps

Fiber bundles with bundle maps between them form a concrete category.

The category Cat consists of all small categories, with functors between them as morphisms.

## 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 and is denoted Cop.

### Product categories

If C and D are categories, one can form the product categoryC × 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 : ab is called

• a monomorphism (or monic) if it is left-cancellable, i.e. fg1 = fg2 implies g1 = g2 for all morphisms g1, g2 : xa.
• an epimorphism (or epic) if it is right-cancellable, i.e. g1f = g2f implies g1 = g2 for all morphisms g1, g2 : bx.
• 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 : ba with fg = 1b.
• a section if it has a left inverse, i.e. if there exists a morphism g : ba with gf = 1a.
• an isomorphism if it has an inverse, i.e. if there exists a morphism g : ba with fg = 1b and gf = 1a.
• 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:

• 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.

## Types of categories

• In many categories, e.g. Ab or VectK, the hom-sets hom(a, b) are not just sets but actually abelian groups, and the composition of morphisms is compatible with these group structures; i.e. is bilinear. Such a category is called preadditive. If, furthermore, the category has all finite products and coproducts, it is called an additive category. If all morphisms have a kernel and a cokernel, and all epimorphisms are cokernels and all monomorphisms are kernels, then we speak of an abelian category. A typical example of an abelian category is the category of abelian groups.
• A category is called complete if all small 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. Examples include Set and CPO, the category of complete partial orders with Scott-continuous functions.
• 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.

## Notes

1. Barr & Wells 2005 , Chapter 1
2. Some authors write Mor(a, b) or simply C(a, b) instead.

## Related Research Articles

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows. A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.

In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.

In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

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 desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very stable categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named after Niels Henrik Abel.

In mathematics, specifically category theory, adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems, such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.

In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-categoryC is a category such that every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas:

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation .

In category theory, an epimorphism is a morphism f : XY that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: YZ,

In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts.

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms. In an enriched category, the set of morphisms associated with every pair of objects is replaced by an object in some fixed monoidal category of "hom-objects". In order to emulate the (associative) composition of morphisms in an ordinary category, the hom-category must have a means of composing hom-objects in an associative manner: that is, there must be a binary operation on objects giving us at least the structure of a monoidal category, though in some contexts the operation may also need to be commutative and perhaps also to have a right adjoint.

In mathematics, a monoidal category is a category equipped with a bifunctor

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". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

In category theory, a coequalizer is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer.

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.

This is a glossary of properties and concepts in category theory in mathematics.

In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if f : XY and g : YX are morphisms whose composition fog : YY is the identity morphism on Y, then g is a section of f, and f is a retraction of g.

In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings and whose morphisms are ring homomorphisms. Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper.

In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.

## References

• Adámek, Jiří; Herrlich, Horst; Strecker, George E. (1990), Abstract and Concrete Categories (PDF), Wiley, ISBN   0-471-60922-6 (now free on-line edition, GNU FDL).
• Asperti, Andrea; Longo, Giuseppe (1991), , MIT Press, ISBN   0-262-01125-5 .
• Awodey, Steve (2006), Category theory, Oxford logic guides, 49, Oxford University Press, ISBN   978-0-19-856861-2 .
• Barr, Michael; Wells, Charles (2005), Toposes, Triples and Theories, Reprints in Theory and Applications of Categories, 12 (revised ed.), MR   2178101 .
• Borceux, Francis (1994), "Handbook of Categorical Algebra", Encyclopedia of Mathematics and its Applications, 50–52, Cambridge: Cambridge University Press, ISBN   0-521-06119-9 .
• "Category", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
• Herrlich, Horst; Strecker, George E. (2007), Category Theory, Heldermann Verlag, ISBN   978-3-88538-001-6 .
• Jacobson, Nathan (2009), Basic algebra (2nd ed.), Dover, ISBN   978-0-486-47187-7 .
• Lawvere, William; Schanuel, Steve (1997), Conceptual Mathematics: A First Introduction to Categories, Cambridge University Press, ISBN   0-521-47249-0 .
• Mac Lane, Saunders (1998), Categories for the Working Mathematician , Graduate Texts in Mathematics, 5 (2nd ed.), Springer-Verlag, ISBN   0-387-98403-8 .
• Marquis, Jean-Pierre (2006), "Category Theory", in Zalta, Edward N. (ed.), Stanford Encyclopedia of Philosophy .
• Sica, Giandomenico (2006), What is category theory?, Advanced studies in mathematics and logic, 3, Polimetrica, ISBN   978-88-7699-031-1 .
• category in nLab