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 (for example) 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.
- Consider the category of finite-dimensional real vector spaces, and the category of all real matrices (the latter category is explained in the article on additive categories). 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 (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.
- Any category is equivalent to its skeleton.
Read more about this topic: Equivalence Of Categories
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