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". In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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.

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 Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the

An equivalence of categories consists of a functor between the involved categories, which is required to have an "inverse" functor. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories 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. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective Instead it is sufficient that each object be naturally isomorphic to its image under this composition. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal 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. In Category theory, two categories C and D are isomorphic if there exist Functors F: C &rarr D and G

Contents 1 Definition 2 Equivalent characterizations 3 Examples 4 Properties

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 ε: FG→I_D and η : 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. In Category theory, a faithful functor (resp a full functor) is a Functor which is Injective (resp 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. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Category theory, a faithful functor (resp a full functor) is a Functor which is Injective (resp 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. In Category theory, a Functor FC\to D is essentially surjective if each object d of D is isomorphic 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. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. 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 (unfortunately this conflicts with terminology from homotopy 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:

• There are natural isomorphisms 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 as the two objects therein are not isomorphic.

• 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). 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 (called vectors) that informally speaking may be scaled and added In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, specifically in Category theory, an additive category is a Preadditive category C such that any finitely many objects A 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. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property The functor G associates to every commutative ring its spectrum, the scheme defined by the prime ideals of the ring. In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of 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 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. For functional analysis as used in psychology see the Functional analysis (psychology article C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space 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. iDEAL is an Internet payment method in The Netherlands, based on online banking This is the Gelfand representation. In Mathematics, the Gelfand representation in Functional analysis (named after I

• In lattice theory, there are a number of dualities, based on representation theorems that connect certain classes of lattices to classes of topological spaces. In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of 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. In Mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is Isomorphic to a Field of sets. In Mathematics, there is an ample supply of categorical dualities between certain categories of Topological spaces and categories of Partially ordered Each Boolean algebra B is mapped to a specific topology on the set of ultrafilters of B. In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' 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 Mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is Isomorphic to a Field of sets.
• In pointless topology the category of spatial locales is known to be equivalent to the dual of the category of sober spaces. In Mathematics, pointless topology (also called point-free or pointfree topology is an approach to Topology which avoids the mentioning of points

• Any category is equivalent to its skeleton. In Mathematics, a skeleton of a category is a Subcategory which roughly speaking does not contain any extraneous Isomorphisms In a certain

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. ↔ 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 &rarr Y which In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective
• the functor H : I → C has limit (or colimit) l if and only if the functor FH : I → D has limit (or colimit) Fl. 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 This can be applied to equalizers, products and coproducts among others. In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological Applying it to kernels and cokernels, we see that the equivalence F is an exact functor. In Category theory and its applications to other branches of Mathematics, kernels are a generalization of the kernels of Group homomorphisms and the kernels In Homological algebra, an exact functor is a Functor, from some category to another which preserves Exact sequences Exact functors are very
• C is a cartesian closed category (or a topos) if and only if D is cartesian closed (or a topos). 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 In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets

Dualities "turn all concepts around": they turn initial objects into terminal objects, monomorphisms into epimorphisms, kernels into cokernels, limits into colimits etc.

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. In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category In Mathematics, specifically in Category theory, an additive category is a Preadditive category C such that any finitely many objects A In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category 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. )

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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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 Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously )

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