Equivalence of categories

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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 dual of another category then one speaks of a duality of categories.

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.

Table of contents

1 Definition
2 Equivalent Characterizations
3 Examples
4 Properties

Definition

Formally, given two categories C and D, an equivalence of categories is a functor F : C -> D such that there is a functor G : D -> C with the composition FG naturally isomorphic to I_D, and GF naturally isomorphic to I_C. Here I_D denotes the identity functor D -> D that assigns every object and every morphism to itself.

In this situation, we say that the categories C and D are equivalent. If F and G are contravariant functors, then one speaks instead of a ''duality of categories''.

Equivalent Characterizations

The above definition is probably the easiest one of many equivalent statements, some of which are listed below.

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

full, i.e. for any two objects c₁ and c₂ of C, the map Mor_C(c₁,c₂) -> Mor_D(Fc₁,Fc₂) induced by F is surjective;
faithful, i.e. for any two objects c₁ and c₂ of C, the map Mor_C(c₁,c₂) -> Mor_D(Fc₁,Fc₂) induced by F is injective; and
dense, 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.

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:

FG is naturally isomorphic to I_D and GF is naturally isomorphic to I_C
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".

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 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 dense.
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.

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), iff 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), iff Fα is a monomorphihsm (or epimorphism, or isomorphism) in D.
the functor H : I -> C has limit (or colimit) l iff the functor FH : I -> D has limit (or colimit) Fl. This can be applied to equalizers, productss and coproducts among others. Applying it to kernelss and cokernels, we see that the equivalence F is an exact functor.
C is a cartesian closed category (or a topos) iff 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 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.)

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.)