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Category theory is a general theory ofmathematical structures and their relations. It was introduced bySamuel Eilenberg andSaunders Mac Lane in the mid-20th century in their foundational work onalgebraic topology.[1] Category theory is used in most areas of mathematics. In particular, many constructions of newmathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples includequotient spaces,direct products, completion, andduality.
Many areas ofcomputer science also rely on category theory, such asfunctional programming andsemantics.
Acategory is formed by two sorts ofobjects: theobjects of the category, and themorphisms, that relate two objects called thesource and thetarget of the morphism. A morphism is often represented by an arrow from its source to its target (see the figure). Morphisms can be composed if the target of the first morphism equals the source of the second one. Morphism composition has similar properties asfunction composition (associativity and existence of anidentity morphism for each object). Morphisms are often some sort offunctions, but this is not always the case. For example, amonoid may be viewed as a category with a single object, whose morphisms are the elements of the monoid.
The second fundamental concept of category theory is the concept of afunctor, which plays the role of a morphism between two categories and: it maps objects of to objects of and morphisms of to morphisms of in such a way that sources are mapped to sources, and targets are mapped to targets (or, in the case of acontravariant functor, sources are mapped to targets andvice-versa). A third fundamental concept is anatural transformation that may be viewed as a morphism of functors.
Acategory consists of the following three mathematical entities:
Each morphism has asource object andtarget object.
The expression would be verbally stated as " is a morphism froma tob".
The expression – alternatively expressed as,, or – denotes thehom-class of all morphisms from to.[a]
From the axioms, it can be proved that there is exactly one identity morphism for every object.
Relations among morphisms (such asfg =h) are often depicted usingcommutative diagrams, with "points" (corners) representing objects and "arrows" representing morphisms.
Morphisms can have any of the following properties. A morphismf :a →b is:
Every retraction is an epimorphism, and every section is a monomorphism. Furthermore, the following three statements are equivalent:
Functors are structure-preserving maps between categories. They can be thought of as morphisms in the category of all (small) categories.
A (covariant) functorF from a categoryC to a categoryD, writtenF :C →D, consists of:
such that the following two properties hold:
Acontravariant functorF:C →D is like a covariant functor, except that it "turns morphisms around" ("reverses all the arrows"). More specifically, every morphismf :x →y inC must be assigned to a morphismF(f) :F(y) →F(x) inD. In other words, a contravariant functor acts as a covariant functor from theopposite categoryCop toD.
Anatural transformation is a relation between two functors. Functors often describe "natural constructions" and natural transformations then describe "natural homomorphisms" between two such constructions. Sometimes two quite different constructions yield "the same" result; this is expressed by a natural isomorphism between the two functors.
IfF andG are (covariant) functors between the categoriesC andD, then a natural transformationη fromF toG associates to every objectX inC a morphismηX :F(X) →G(X) inD such that for every morphismf :X →Y inC, we haveηY ∘F(f) =G(f) ∘ηX; this means that the following diagram iscommutative:
The two functorsF andG are callednaturally isomorphic if there exists a natural transformation fromF toG such thatηX is an isomorphism for every objectX inC.
Using the language of category theory, many areas of mathematical study can be categorized. Categories include sets, groups and topologies.
Each category is distinguished by properties that all its objects have in common, such as theempty set or theproduct of two topologies, yet in the definition of a category, objects are considered atomic, i.e., wedo not know whether an objectA is a set, a topology, or any other abstract concept. Hence, the challenge is to define special objects without referring to the internal structure of those objects. To define the empty set without referring to elements, or the product topology without referring to open sets, one can characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories. Thus, the task is to finduniversal properties that uniquely determine the objects of interest.
Numerous important constructions can be described in a purely categorical way if thecategory limit can be developed and dualized to yield the notion of acolimit.
It is a natural question to ask: under which conditions can two categories be consideredessentially the same, in the sense that theorems about one category can readily be transformed into theorems about the other category? The major tool one employs to describe such a situation is calledequivalence of categories, which is given by appropriate functors between two categories. Categorical equivalence has foundnumerous applications in mathematics.
The definitions of categories and functors provide only the very basics of categorical algebra; additional important topics are listed below. Although there are strong interrelations between all of these topics, the given order can be considered as a guideline for further reading.
Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can be situated into the context ofhigher-dimensional categories. Briefly, if we consider a morphism between two objects as a "process taking us from one object to another", then higher-dimensional categories allow us to profitably generalize this by considering "higher-dimensional processes".
For example, a (strict)2-category is a category together with "morphisms between morphisms", i.e., processes which allow us to transform one morphism into another. We can then "compose" these "bimorphisms" both horizontally and vertically, and we require a 2-dimensional "exchange law" to hold, relating the two composition laws. In this context, the standard example isCat, the 2-category of all (small) categories, and in this example, bimorphisms of morphisms are simplynatural transformations of morphisms in the usual sense. Another basic example is to consider a 2-category with a single object; these are essentiallymonoidal categories.Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative "up to" an isomorphism.
This process can be extended for allnatural numbersn, and these are calledn-categories. There is even a notion ofω-category corresponding to theordinal numberω.
Higher-dimensional categories are part of the broader mathematical field ofhigher-dimensional algebra, a concept introduced byRonald Brown. For a conversational introduction to these ideas, seeJohn Baez, 'A Tale ofn-categories' (1996).
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It should be observed first that the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of a natural transformation [...]
Whilst specific examples of functors and natural transformations had been given bySamuel Eilenberg andSaunders Mac Lane in a 1942 paper ongroup theory,[3] these concepts were introduced in a more general sense, together with the additional notion of categories, in a 1945 paper by the same authors[2] (who discussed applications of category theory to the field ofalgebraic topology).[4] Their work was an important part of the transition from intuitive and geometrichomology tohomological algebra, Eilenberg and Mac Lane later writing that their goal was to understand natural transformations, which first required the definition of functors, then categories.
Stanislaw Ulam, and some writing on his behalf, have claimed that related ideas were current in the late 1930s in Poland.[citation needed] Eilenberg was Polish, and studied mathematics in Poland in the 1930s.[5] Category theory is also, in some sense, a continuation of the work ofEmmy Noether (one of Mac Lane's teachers) in formalizing abstract processes;[6] Noether realized that understanding a type of mathematical structure requires understanding the processes that preserve that structure (homomorphisms).[citation needed] Eilenberg and Mac Lane introduced categories for understanding and formalizing the processes (functors) that relatetopological structures to algebraic structures (topological invariants) that characterize them.
Category theory was originally introduced for the need ofhomological algebra, and widely extended for the need of modernalgebraic geometry (scheme theory). Category theory may be viewed as an extension ofuniversal algebra, as the latter studiesalgebraic structures, and the former applies to any kind ofmathematical structure and studies also the relationships between structures of different nature. For this reason, it is used throughout mathematics. Applications tomathematical logic andsemantics (categorical abstract machine) came later.
Certain categories calledtopoi (singulartopos) can even serve as an alternative toaxiomatic set theory as a foundation of mathematics. A topos can also be considered as a specific type of category with two additional topos axioms. These foundational applications of category theory have been worked out in fair detail as a basis for, and justification of,constructive mathematics.Topos theory is a form of abstractsheaf theory, with geometric origins, and leads to ideas such aspointless topology.
Categorical logic is now a well-defined field based ontype theory forintuitionistic logics, with applications infunctional programming anddomain theory, where acartesian closed category is taken as a non-syntactic description of alambda calculus. At the very least, category theoretic language clarifies what exactly these related areas have in common (in someabstract sense).
Category theory has been applied in other fields as well, seeapplied category theory. For example,John Baez has shown a link betweenFeynman diagrams inphysics and monoidal categories.[7] Another application of category theory, more specifically topos theory, has been made in mathematical music theory, see for example the bookThe Topos of Music, Geometric Logic of Concepts, Theory, and Performance byGuerino Mazzola.
More recent efforts to introduce undergraduates to categories as a foundation for mathematics include those ofLawvere & Rosebrugh (2003),[8] and Lawvere andStephen Schanuel (1997).
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