Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology . Category theory is used in almost all areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces , direct products , completion, and duality .
41-556: Ross Howard Street (born 29 September 1945, Sydney) is an Australian mathematician specialising in category theory . Street completed his undergraduate and postgraduate study at the University of Sydney , where his dissertation advisor was Max Kelly . He is an emeritus professor of mathematics at Macquarie University , a fellow of the Australian Mathematical Society (1995), and was elected Fellow of
82-653: A covariant functor on the opposite category C o p {\displaystyle C^{\mathrm {op} }} . Some authors prefer to write all expressions covariantly. That is, instead of saying F : C → D {\displaystyle F\colon C\to D} is a contravariant functor, they simply write F : C o p → D {\displaystyle F\colon C^{\mathrm {op} }\to D} (or sometimes F : C → D o p {\displaystyle F\colon C\to D^{\mathrm {op} }} ) and call it
123-475: A functor is a mapping between categories . Functors were first considered in algebraic topology , where algebraic objects (such as the fundamental group ) 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
164-508: 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 is Cat , the 2-category of all (small) categories, and in this example, bimorphisms of morphisms are simply natural transformations of morphisms in
205-820: A category, and similarly for D {\displaystyle D} , F o p {\displaystyle F^{\mathrm {op} }} is distinguished from F {\displaystyle F} . For example, when composing F : C 0 → C 1 {\displaystyle F\colon C_{0}\to C_{1}} with G : C 1 o p → C 2 {\displaystyle G\colon C_{1}^{\mathrm {op} }\to C_{2}} , one should use either G ∘ F o p {\displaystyle G\circ F^{\mathrm {op} }} or G o p ∘ F {\displaystyle G^{\mathrm {op} }\circ F} . Note that, following
246-553: A category: the functor category . Morphisms in this category are natural transformations between functors. Functors are often defined by universal properties ; examples are the tensor product , the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits. The concepts of limit and colimit generalize several of the above. Universal constructions often give rise to pairs of adjoint functors . Functors sometimes appear in functional programming . For instance,
287-424: 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 abstract sheaf theory , with geometric origins, and leads to ideas such as pointless topology . Categorical logic
328-405: A functor and of a natural transformation [...] Whilst specific examples of functors and natural transformations had been given by Samuel Eilenberg and Saunders Mac Lane in a 1942 paper on group theory , these concepts were introduced in a more general sense, together with the additional notion of categories, in a 1945 paper by the same authors (who discussed applications of category theory to
369-411: A functor. Contravariant functors are also occasionally called cofunctors . There is a convention which refers to "vectors"—i.e., vector fields , elements of the space of sections Γ ( T M ) {\displaystyle \Gamma (TM)} of a tangent bundle T M {\displaystyle TM} —as "contravariant" and to "covectors"—i.e., 1-forms , elements of
410-615: A link between Feynman diagrams in physics and monoidal categories. Another application of category theory, more specifically topos theory, has been made in mathematical music theory, see for example the book The Topos of Music, Geometric Logic of Concepts, Theory, and Performance by Guerino Mazzola . More recent efforts to introduce undergraduates to categories as a foundation for mathematics include those of William Lawvere and Rosebrugh (2003) and Lawvere and Stephen Schanuel (1997) and Mirroslav Yotov (2012). Contravariant functor In mathematics , specifically category theory ,
451-454: A natural transformation η from F to G associates to every object X in C a morphism η X : F ( X ) → G ( X ) in D such that for every morphism f : X → Y in C , we have η Y ∘ F ( f ) = G ( f ) ∘ η X ; this means that the following diagram is commutative : The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that η X
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#1732776879424492-624: A way that sources are mapped to sources, and targets are mapped to targets (or, in the case of a contravariant functor , sources are mapped to targets and vice-versa ). A third fundamental concept is a natural transformation that may be viewed as a morphism of functors. A category C {\displaystyle {\mathcal {C}}} consists of the following three mathematical entities: Relations among morphisms (such as fg = h ) are often depicted using commutative diagrams , with "points" (corners) representing objects and "arrows" representing morphisms. Morphisms can have any of
533-407: Is a natural example; it is contravariant in one argument, covariant in the other. A multifunctor is a generalization of the functor concept to n variables. So, for example, a bifunctor is a multifunctor with n = 2 . Two important consequences of the functor axioms are: One can compose functors, i.e. if F is a functor from A to B and G is a functor from B to C then one can form
574-402: 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. If F and G are (covariant) functors between the categories C and D , then
615-418: 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,
656-444: Is an isomorphism for every object X in C . 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 the empty set or the product of two topologies , yet in the definition of a category, objects are considered atomic, i.e., we do not know whether an object A
697-476: Is applied. The words category and functor were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap , respectively. The latter used functor in a linguistic context; see function word . Let C and D be categories . A functor F from C to D is a mapping that That is, functors must preserve identity morphisms and composition of morphisms. There are many constructions in mathematics that would be functors but for
738-473: Is even a notion of ω-category corresponding to the ordinal number ω . Higher-dimensional categories are part of the broader mathematical field of higher-dimensional algebra , a concept introduced by Ronald Brown . For a conversational introduction to these ideas, see John Baez, 'A Tale of n -categories' (1996). It should be observed first that the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of
779-869: Is not always the case. For example, a monoid 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 a functor , which plays the role of a morphism between two categories C 1 {\displaystyle {\mathcal {C}}_{1}} and C 2 {\displaystyle {\mathcal {C}}_{2}} : it maps objects of C 1 {\displaystyle {\mathcal {C}}_{1}} to objects of C 2 {\displaystyle {\mathcal {C}}_{2}} and morphisms of C 1 {\displaystyle {\mathcal {C}}_{1}} to morphisms of C 2 {\displaystyle {\mathcal {C}}_{2}} in such
820-505: Is now a well-defined field based on type theory for intuitionistic logics , with applications in functional programming and domain theory , where a cartesian closed category is taken as a non-syntactic description of a lambda calculus . At the very least, category theoretic language clarifies what exactly these related areas have in common (in some abstract sense). Category theory has been applied in other fields as well, see applied category theory . For example, John Baez has shown
861-553: Is observed that the coordinate transformation symbol Λ i j {\displaystyle \Lambda _{i}^{j}} (representing the matrix Λ T {\displaystyle {\boldsymbol {\Lambda }}^{\textsf {T}}} ) acts on the "covector coordinates" "in the same way" as on the basis vectors: e i = Λ i j e j {\displaystyle \mathbf {e} _{i}=\Lambda _{i}^{j}\mathbf {e} _{j}} —whereas it acts "in
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#1732776879424902-504: The objects of the category, and the morphisms , which relate two objects called the source and the target of the morphism. Metaphorically, a morphism is an arrow that maps its source to its target. Morphisms can be composed if the target of the first morphism equals the source of the second one. Morphism composition has similar properties as function composition ( associativity and existence of an identity morphism for each object). Morphisms are often some sort of functions , but this
943-490: The opposite categories to C {\displaystyle C} and D {\displaystyle D} . By definition, F o p {\displaystyle F^{\mathrm {op} }} maps objects and morphisms in the identical way as does F {\displaystyle F} . Since C o p {\displaystyle C^{\mathrm {op} }} does not coincide with C {\displaystyle C} as
984-414: The opposite functor F o p : C o p → D o p {\displaystyle F^{\mathrm {op} }\colon C^{\mathrm {op} }\to D^{\mathrm {op} }} , where C o p {\displaystyle C^{\mathrm {op} }} and D o p {\displaystyle D^{\mathrm {op} }} are
1025-695: The Australian Academy of Science in 1989. He was awarded the Edgeworth David Medal of the Royal Society of New South Wales in 1977, and the Australian Mathematical Society's George Szekeres Medal in 2012. Category theory Many areas of computer science also rely on category theory, such as functional programming and semantics . A category is formed by two sorts of objects :
1066-446: The composite functor G ∘ F from A to C . Composition of functors is associative where defined. Identity of composition of functors is the identity functor. This shows that functors can be considered as morphisms in categories of categories, for example in the category of small categories . A small category with a single object is the same thing as a monoid : the morphisms of a one-object category can be thought of as elements of
1107-424: The fact that they "turn morphisms around" and "reverse composition". We then define a contravariant functor F from C to D as a mapping that Variance of functor (composite) Note that contravariant functors reverse the direction of composition. Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as
1148-415: The field of algebraic topology ). Their work was an important part of the transition from intuitive and geometric homology to homological 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
1189-468: The following properties. A morphism f : a → b is a: 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 ) functor F from a category C to a category D , written F : C → D , consists of: such that
1230-457: The following two properties hold: A contravariant functor F : C → D is like a covariant functor, except that it "turns morphisms around" ("reverses all the arrows"). More specifically, every morphism f : x → y in C must be assigned to a morphism F ( f ) : F ( y ) → F ( x ) in D . In other words, a contravariant functor acts as a covariant functor from the opposite category C to D . A natural transformation
1271-400: The former applies to any kind of mathematical structure and studies also the relationships between structures of different nature. For this reason, it is used throughout mathematics. Applications to mathematical logic and semantics ( categorical abstract machine ) came later. Certain categories called topoi (singular topos ) can even serve as an alternative to axiomatic set theory as
Ross Street - Misplaced Pages Continue
1312-497: 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 of higher-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,
1353-534: The late 1930s in Poland. Eilenberg was Polish, and studied mathematics in Poland in the 1930s. Category theory is also, in some sense, a continuation of the work of Emmy Noether (one of Mac Lane's teachers) in formalizing abstract processes; Noether realized that understanding a type of mathematical structure requires understanding the processes that preserve that structure ( homomorphisms ). Eilenberg and Mac Lane introduced categories for understanding and formalizing
1394-407: The monoid, and composition in the category is thought of as the monoid operation. Functors between one-object categories correspond to monoid homomorphisms . So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object. Let C and D be categories. The collection of all functors from C to D forms the objects of
1435-676: The opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors: e i = Λ j i e j {\displaystyle \mathbf {e} ^{i}=\Lambda _{j}^{i}\mathbf {e} ^{j}} ). This terminology is contrary to the one used in category theory because it is the covectors that have pullbacks in general and are thus contravariant , whereas vectors in general are covariant since they can be pushed forward . See also Covariance and contravariance of vectors . Every functor F : C → D {\displaystyle F\colon C\to D} induces
1476-465: The other category? The major tool one employs to describe such a situation is called equivalence of categories , which is given by appropriate functors between two categories. Categorical equivalence has found numerous 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,
1517-425: The processes ( functors ) that relate topological structures to algebraic structures ( topological invariants ) that characterize them. Category theory was originally introduced for the need of homological algebra , and widely extended for the need of modern algebraic geometry ( scheme theory ). Category theory may be viewed as an extension of universal algebra , as the latter studies algebraic structures , and
1558-481: The property of opposite category , ( F o p ) o p = F {\displaystyle \left(F^{\mathrm {op} }\right)^{\mathrm {op} }=F} . A bifunctor (also known as a binary functor ) is a functor whose domain is a product category . For example, the Hom functor is of the type C × C → Set . It can be seen as a functor in two arguments. The Hom functor
1599-1196: The space of sections Γ ( T ∗ M ) {\displaystyle \Gamma {\mathord {\left(T^{*}M\right)}}} of a cotangent bundle T ∗ M {\displaystyle T^{*}M} —as "covariant". This terminology originates in physics, and its rationale has to do with the position of the indices ("upstairs" and "downstairs") in expressions such as x ′ i = Λ j i x j {\displaystyle {x'}^{\,i}=\Lambda _{j}^{i}x^{j}} for x ′ = Λ x {\displaystyle \mathbf {x} '={\boldsymbol {\Lambda }}\mathbf {x} } or ω i ′ = Λ i j ω j {\displaystyle \omega '_{i}=\Lambda _{i}^{j}\omega _{j}} for ω ′ = ω Λ T . {\displaystyle {\boldsymbol {\omega }}'={\boldsymbol {\omega }}{\boldsymbol {\Lambda }}^{\textsf {T}}.} In this formalism it
1640-471: The task is to find universal properties that uniquely determine the objects of interest. Numerous important constructions can be described in a purely categorical way if the category limit can be developed and dualized to yield the notion of a colimit . It is a natural question to ask: under which conditions can two categories be considered essentially the same , in the sense that theorems about one category can readily be transformed into theorems about
1681-415: The usual sense. Another basic example is to consider a 2-category with a single object; these are essentially monoidal 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 all natural numbers n , and these are called n -categories . There