Isomorphisms

This convention is not universally followed, and authors who wish to distinguish between unnatural isomorphisms and natural isomorphisms will generally explicitly state the distinction.

Additionally, every pair of adjoint functors comes equipped with two natural transformations (generally not isomorphisms) called the unit and counit.

Additivity If C and D are preadditive categories and F : C ← D is an additive functor with a right adjoint G : C → D, then G is also an additive functor and the hom-set bijections : are, in fact, isomorphisms of abelian groups.

Bijections and category theory Bijections are precisely the isomorphisms in the category Set of sets and set functions.

Equivalences of categories If a functor F: C←D is one half of an equivalence of categories then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms.

Example: dual of a finite-dimensional vector space Every finite-dimensional vector space is isomorphic to its dual space, but there may be many different isomorphisms between the two spaces.

For a detailed discussion of relational homomorphisms and isomorphisms see.

For infinite-dimensional vector spaces, inequivalent topologies lead to inequivalent notions of tensor, and these various isomorphisms may or may not hold depending on what exactly is meant by a tensor (see topological tensor product ).

Homeomorphisms are the isomorphisms in the category of topological spaces that is, they are the mappings that preserve all the topological properties of a given space.

However, the bijections are not always the isomorphisms for more complex categories.

Logarithmic functions are the only continuous isomorphisms between these groups.

One would expect that all isomorphisms and all compositions of embeddings are embeddings, and that all embeddings are monomorphisms.

Properties of the category of sets The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps.

Research Announcement: "Deduction-preserving 'Recursive Isomorphisms' between Theories" (with Marian Boykan Pour-El), Bulletin of the American Mathematical Society, 73:145-148.

Some example isomorphisms: * Every regular G action is isomorphic to the action of G on G given by left multiplication.

Some notable special monotone functions are order embeddings (functions for which x ≤ y if and only if f(x) ≤ f(y)) and order isomorphisms ( surjective order embeddings).

The above statements about isomorphisms for regular, free and transitive actions are no longer valid for continuous group actions.

The indiscrete topology is also known as the biggest or chaotic topology, and it is generated by the pretopology which has only isomorphisms for covering families.

The result has been extended by Amir see D. Amir, "On isomorphisms of continuous function spaces".

The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore isomorphic to each other via many possible isomorphisms.