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 such an equivalence usually means to discover strong similarities between mathematical structures that formerly were considered to be unrelated or where the relation was not understood properly. The gain of this usually is a better understanding of the nature of the considered objects and the possibility to translate theorems between different kinds of mathematical structures. If a category is equivalent to the opposite (or dual) of another category then one speaks of a duality of categories, and says that the two categories are dually equivalent. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... 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...

An equivalence of categories consists of a functor between the involved categories, which is required to have an "inverse" functor. However, in contrast to the situation common for isomorphisms in an algebraic setting, the composition of the functor and its "inverse" is not necessarily the identity mapping. Instead it is sufficient that each object be naturally isomorphic to its image under this composition. Thus one may describe the functors as being "inverse up to isomorphism". There is indeed a concept of isomorphism of categories where a strict form of inverse functor is required, but this is of much less practical use than the equivalence concept. Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... 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 category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i. ...

Definition

Formally, given two categories C and D, an equivalence of categories consists of a functor F : C → D, a functor G : D → C, and two natural isomorphisms η: FG→I_D and ε: I_C→GF. Here FG: D→D and GF: C→C, denote the respective compositions of F and G, and I_C: C→C and I_D: D→D denote the identity functors on C and D, assigning each object and morphism to itself. If F and G are contravariant functors one speaks of a duality of categories instead.

One often does not specify all the above data. For instance, we say that the categories C and D are equivalent (respectively dually equivalent) if there exists an equivalence (respectively duality) between them. Furthermore, we say that F "is" an equivalence of categories if an inverse functor G and natural isomorphisms as above exist. Note however that knowledge of F is usually not enough to reconstruct G and the natural isomorphisms: there may be many choices (see example below).

Equivalent characterizations

One can show that a functor F : C → D yields an equivalence of categories if and only if it is:

• full, i.e. for any two objects c₁ and c₂ of C, the map Hom_C(c₁,c₂) → Hom_D(Fc₁,Fc₂) induced by F is surjective;
• faithful, i.e. for any two objects c₁ and c₂ of C, the map Hom_C(c₁,c₂) → Hom_D(Fc₁,Fc₂) induced by F is injective; and
• essentially surjective, i.e. each object d in D is isomorphic to an object of the form Fc, for c in C.

This is a quite useful and commonly applied criterion, because one does not have to explicitly construct the "inverse" G and the natural isomorphisms between FG, GF and the identity functors. On the other hand, though the above properties guarantee the existence of a categorical equivalence (given a sufficiently strong version of the axiom of choice in the underlying set theory), the missing data is not completely specified, and often there are many choices. It is a good idea to specify the missing constructions explicitly whenever possible. Due to this circumstance, a functor with these properties is sometimes called a weak equivalence of categories. In category theory, a full functor is a functor which is surjective when restricted to each set of morphisms with a given source and target. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In category theory, a faithful functor is a functor which is injective when restricted to each set of morphisms with a given source and target. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In category theory, a functor is essentially surjective when each object of is isomorphic to an object of the form for some object of . ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

There is also a close relation to the concept of adjoint functors. The following statements are equivalent for functors F : C → D and G : D → C: The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...

• There are natural transformations from FG to I_D and I_C to GF called the co-unit and unit resp.
• F is a left adjoint of G and both functors are full and faithful.
• F is a right adjoint of G and both functors are full and faithful.

One may therefore view an adjointness relation between two functors as a "very weak form of equivalence". Assuming that the natural transformations for the adjunctions are given, all of these formulations allow for an explicit construction of the necessary data, and no choice principles are needed. The key property that one has to prove here is that the counit of an adjunction is an isomorphism if and only if the right adjoint is a full and faithful functor.

Examples

• Consider the category C having a single object c and a single morphism 1_c, and the category D with two objects d₁, d₂ and four morphisms: two identity morphisms 1_d1, 1_d2 and two isomorphisms α:d₁→d₂ and β:d₂→d₁. The categories C and D are equivalent; we can (for example) have F map c to d₁ and G map both objects of D to c and all morphisms to 1_c.

• By contrast, the category C with a single object and a single morphism is not equivalent to the category E with two objects and only two identity morphisms.

• Consider a category C with one object c, and two morphisms 1, f: c→c. Let 1 be the identity morphism on c and set f o f = 1. Of course, C is equivalent to itself, which can be shown by taking 1 in place of the required natural isomorphisms between the functor I_C and itself. However, it is also true that f yields a natural isomorphism from I_C to itself. Hence, given the information that the identity functors form an equivalence of categories, in this example one still can choose between two natural isomorphisms for each direction.

• Consider the category C of finite-dimensional real vector spaces, and the category D = Mat(R) of all real matrices (the latter category is explained in the article on additive categories). Then C and D are equivalent: The functor G : D → C which maps the object A_n of D to the vector space Rⁿ and the matrices in D to the corresponding linear maps is full, faithful and essentially surjective.

• One of the central themes of algebraic geometry is the duality of the category of affine schemes and the category of commutative rings. The functor G associates to every commutative ring its spectrum, the scheme defined by the prime ideals of the ring. Its adjoint F associates to every affine scheme its ring of global sections.

• In functional analysis the category of commutative C*-algebras with identity is contravariantly equivalent to the category of compact Hausdorff spaces. Under this duality, every compact Hausdorff space X is associated with the algebra of continuous complex-valued functions on X, and every commutative C*-algebra is associated with the space of its maximal ideals. This is the Gelfand representation.

• In lattice theory, there are a number of dualities, based on representation theorems that connect certain classes of lattices to classes of topological spaces. Probably the most well-known theorem of this kind is Stone's representation theorem for Boolean algebras, which is a special instance within the general scheme of Stone duality. Each Boolean algebra B is mapped to a specific topology on the set of ultrafilters of B. Conversely, for any topology the clopen (i.e. closed and open) subsets yield a Boolean algebra. One obtains a duality between the category of Boolean algebras (with their homomorphisms) and Stone spaces (with continuous mappings).

• In pointless topology the category of spatial locales is known to be equivalent to the dual of the category of sober spaces.

• Any category is equivalent to its skeleton.

In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... 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... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... C*-algebras are an important area of research in functional analysis. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... iDEAL is an Internet payment method in The Netherlands, based on online banking. ... In mathematics, the Gelfand representation in functional analysis allows a complete characterisation of commutative C*-algebras as algebras of continuous complex-valued functions. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... In mathematics, Stones representation theorem for Boolean algebras, named in honor of Marshall H. Stone, is the duality between the category of Boolean algebras and the category of Stone spaces, i. ... In mathematics, especially in topology and order theory, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... In mathematics, Stones representation theorem for Boolean algebras, named in honor of Marshall H. Stone, is the duality between the category of Boolean algebras and the category of Boolean spaces, i. ... Pointless topology is an approach to topology which avoids the mentioning of points. ... In mathematics, a skeleton of a category is a subcategory which, roughly speaking, does not contain any extraneous isomorphisms. ...

Properties

As a rule of thumb, an equivalence of categories preserves all "categorical" concepts and properties. If F : C → D is an equivalence, then the following statements are all true:

• the object c of C is an initial object (or terminal object, or zero object), if and only if Fc is an initial object (or terminal object, or zero object) of D
• the morphism α in C is a monomorphism (or epimorphism, or isomorphism), if and only if Fα is a monomorphism (or epimorphism, or isomorphism) in D.
• the functor H : I → C has limit (or colimit) l if and only if the functor FH : I → D has limit (or colimit) Fl. This can be applied to equalizers, products and coproducts among others. Applying it to kernels and cokernels, we see that the equivalence F is an exact functor.
• C is a cartesian closed category (or a topos) if and only if D is cartesian closed (or a topos).

Dualities "turn all concepts around": they turn initial objects into terminal objects, monomorphisms into epimorphisms, kernels into cokernels, limits into colimits etc. In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C... It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C... 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 → Y which is right-cancellable in the following sense: g1 o f = g2 o f implies g1 = g2 for all morphisms g1, g2 : Y → Z. Epimorphisms are analogues of surjective functions, but... 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 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. ... Equalizer can mean: Equalizer, an audio processing tool. ... In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ... 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 homological algebra, an exact functor is one which preserves exact sequences. ... 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. ...

If F : C → D is an equivalence of categories, and G₁ and G₂ are two inverses, then G₁ and G₂ are naturally isomorphic.

If F : C → D is an equivalence of categories, and if C is a preadditive category (or additive category, or abelian category), then D may be turned into a preadditive category (or additive category, or abelian category) in such a way that F becomes an additive functor. On the other hand, any equivalence between additive categories is necessarily additive. (Note that the latter statement is not true for equivalences between preadditive categories.) A preadditive category is a category that is enriched over the monoidal category of abelian groups. ... 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 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. ... A preadditive category is a category that is enriched over the monoidal category of abelian groups. ...

An auto-equivalence of a category C is an equivalence F : C → C. The auto-equivalences of C form a group under composition if we consider two auto-equivalences that are naturally isomorphic to be identical. This group captures the essential "symmetries" of C. (One caveat: if C is not a small category, then the auto-equivalences of C may form a proper class rather than a set.) In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... 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 set can be thought of as any collection of distinct things considered as a whole. ...