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category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a branch of abstract mathematics, an equivalence of categories is a relation between two
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) * Category (Kant) *Categories (Peirce) * ...
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. 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. 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 composite of the functor and its "inverse" is not necessarily the identity mapping. Instead it is sufficient that each object be ''
naturally isomorphic In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
'' 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 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.e. ''FG'' = 1''D'' (the identity functor on ''D'') and ''GF'' ...
where a strict form of inverse functor is required, but this is of much less practical use than the ''equivalence'' concept.


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).


Alternative characterizations

A functor ''F'' : ''C'' → ''D'' yields an equivalence of categories if and only if it is simultaneously: * full, i.e. for any two objects ''c''1 and ''c''2 of ''C'', the map Hom''C''(''c''1,''c''2) → Hom''D''(''Fc''1,''Fc''2) induced by ''F'' is surjective; * faithful, i.e. for any two objects ''c''1 and ''c''2 of ''C'', the map Hom''C''(''c''1,''c''2) → Hom''D''(''Fc''1,''Fc''2) induced by ''F'' is injective; and * essentially surjective (dense), i.e. each object ''d'' in ''D'' is isomorphic to an object of the form ''Fc'', for ''c'' in ''C''.Mac Lane (1998), Theorem IV.4.1 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 mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
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. (Unfortunately this conflicts with terminology from homotopy type theory.) There is also a close relation to the concept of
adjoint functors In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...
F\dashv G, where we say that F:C\rightarrow D is the left adjoint of G:D\rightarrow C, or likewise, ''G'' is the right adjoint of ''F''. Then ''C'' and ''D'' are equivalent (as defined above in that there are natural isomorphisms from ''FG'' to I''D'' and I''C'' to ''GF'') if and only if F\dashv G and both ''F'' and ''G'' are full and faithful. When adjoint functors F\dashv G are not both full and faithful, then we may view their adjointness relation as expressing a "weaker form of equivalence" of categories. 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_, and the category D with two objects d_, d_ and four morphisms: two identity morphisms 1_, 1_ and two isomorphisms \alpha \colon d_ \to d_ and \beta \colon d_ \to d_. The categories C and D are equivalent; we can (for example) have F map c to d_ and G map both objects of D to c and all morphisms to 1_. * 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. The two objects in E are ''not'' isomorphic in that there are no morphisms between them. Thus any functor from C to E will not be essentially surjective. * Consider a category C with one object c, and two morphisms 1_, f \colon c \to c. Let 1_ be the identity morphism on c and set f \circ 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 \mathbf_ and itself. However, it is also true that f yields a natural isomorphism from \mathbf_ 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. * The category of sets and
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is ...
s is equivalent to but not isomorphic with the category of pointed sets and point-preserving maps. * Consider the category C of finite-
dimensional In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
vector spaces, and the category D = \mathrm(\mathbb) of all real matrices (the latter category is explained in the article on additive categories). Then C and D are equivalent: The functor G \colon D \to C which maps the object A_ of D to the vector space \mathbb^ and the matrices in D to the corresponding linear maps is full, faithful and essentially surjective. * One of the central themes of
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
is the duality of the category of
affine scheme In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
s and the category of
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
s. The functor G associates to every commutative ring its
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
, 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*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
s with identity is contravariantly equivalent to the category of
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
s. 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 theorem In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or concrete) structure. Examples Algebra * Cayley's theorem states that every group ...
s 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 In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
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). Another case of Stone duality is Birkhoff's representation theorem stating a duality between finite partial orders and finite distributive lattices. * In pointless topology the category of spatial locales is known to be equivalent to the dual of the category of sober spaces. * For two rings ''R'' and ''S'', the product category ''R''-Mod×''S''-Mod is equivalent to (''R''×''S'')-Mod. * Any category is equivalent to its
skeleton A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...
.


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 In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...
), if and only if ''Fc'' is an initial object (or terminal object, or
zero object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...
) of ''D'' * the morphism α in ''C'' is a
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, 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 Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
(or colimit) ''l'' if and only if the functor ''FH'' : ''I'' → ''D'' has limit (or colimit) ''Fl''. This can be applied to equalizers,
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
s and
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprodu ...
s among others. Applying it to
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
s and
cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the name: ...
s, we see that the equivalence ''F'' is an exact functor. * ''C'' is a
cartesian closed category 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. These categories are particularly important in ma ...
(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. If ''F'' : ''C'' → ''D'' is an equivalence of categories, and ''G''1 and ''G''2 are two inverses of ''F'', then ''G''1 and ''G''2 are naturally isomorphic. If ''F'' : ''C'' → ''D'' is an equivalence of categories, and if ''C'' is a
preadditive category 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-category C is a category such that every hom ...
(or
additive category In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts. Definition A category C is preadditive if all its hom-sets are abelian groups and composition of mor ...
, or
abelian category 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 a ...
), 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 another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every h ...
. 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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
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 Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently ...
rather than a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
.)


See also

* Equivalent definitions of mathematical structures


References

* * * {{DEFAULTSORT:Equivalence Of Categories Adjoint functors Category theory Equivalence (mathematics)