Equivalence of categories

From Academic Kids

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.

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.

Contents

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->ID and ε: 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 morphisms 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 has all of the following three properties:

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

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 ID and GF is naturally isomorphic to IC
  • 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 1c, and the category D with two objects d1, d2 and four morphisms: two identity morphisms 1d1, 1d2 and two isomorphisms α:d1d2 and β:d2d1. 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 1c.
  • 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 IC and itself. However, it is also true that f yields a natural isomorphism from IC 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 : DC which maps the object An of D to the vector space Rn and the matrices in D to the corresponding linear maps is full, faithful and essentially surjective.
  • 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:

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.)

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