Equivalence of categories
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 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 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 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.
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→ID and η : IC→GF. Here FG: D→D and GF: C→C, denote the respective compositions of F and G, and IC: C→C and ID: 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 if there exists an equivalence 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.
Equivalent 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 c1 and c2 of C, the map HomC → HomD induced by F is surjective;
- faithful, i.e. for any two objects c1 and c2 of C, the map HomC → HomD 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.
Due to this circumstance, a functor with these properties is sometimes called a weak equivalence of categories.
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 ID and IC to GF.
- F is a left adjoint of G and both functors are full and faithful.
- G is a right adjoint of F and both functors are full and faithful.
Examples
- Consider the category having a single object and a single morphism, and the category with two objects, and four morphisms: two identity morphisms, and two isomorphisms and. The categories and are equivalent; we can have map to and map both objects of to and all morphisms to.
- By contrast, the category with a single object and a single morphism is not equivalent to the category with two objects and only two identity morphisms as the two objects therein are not isomorphic.
- Consider a category with one object, and two morphisms. Let be the identity morphism on and set. Of course, is equivalent to itself, which can be shown by taking in place of the required natural isomorphisms between the functor and itself. However, it is also true that yields a natural isomorphism from 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 functions is equivalent to but not isomorphic with the category of pointed sets and point-preserving maps.
- Consider the category of finite-dimensional real vector spaces, and the category of all real matrices. Then and are equivalent: The functor which maps the object of to the vector space and the matrices in 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 associates to every commutative ring its spectrum, the scheme defined by the prime ideals of the ring. Its adjoint 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 is associated with the algebra of continuous complex-valued functions on, 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 is mapped to a specific topology on the set of ultrafilters of. Conversely, for any topology the clopen subsets yield a Boolean algebra. One obtains a duality between the category of Boolean algebras and Stone spaces. 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 -Mod.
- Any category is equivalent to its skeleton.
Properties
- the object c of C is an initial object, if and only if Fc is an initial object of D
- the morphism α in C is a monomorphism, if and only if Fα is a monomorphism in D.
- the functor H : I → C has limit l if and only if the functor FH : I → D has limit 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 if and only if D is cartesian closed.
If F : C → D is an equivalence of categories, and G1 and G2 are two inverses of F, then G1 and G2 are naturally isomorphic.
If F : C → D is an equivalence of categories, and if C is a preadditive category, then D may be turned into a preadditive category in such a way that F becomes an additive functor. On the other hand, any equivalence between additive categories is necessarily additive.
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.