Isomorphism

https://en.wikipedia.org/wiki/Isomorphism

An isomorphism is a mapping between two structures of the same type that can be reversed by an inverse mapping. It is a bijection, a structure preserving mapping (thus it can be reveresed).

In every category of algebraic structures, an isomorphism is a homomorphism that is a bijection. An isomorphism of an algebraic structure onto itself is called an automorphism.

Two mathematical structures are isomorphic if an isomorphism exists between them. They have the same properties up to an isomorphism (i.e. disregarding further info such as additional structure, names of objects and similar). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism.

An automorphism is an isomorphism from a structure to itself.

An isomorphism between two structures is a canonical isomorphism if it's the only one (as it is the case for solutions of a universal property), or, if it's more natural (in some sense) than others.

For example, for every prime number p, all fields with p elements are canonically isomorphic, with a unique isomorphism. The isomorphism theorems provide canonical isomorphisms that are not unique.

The term isomorphism is mainly used for algebraic structures. There, mappings are called homomorphisms, and a homomorphism is an isomorphism iff it's bijective.

In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. For example:

• A permutation is an automorphism of a set.

• An isometry is an isomorphism of metric spaces.

• A homeomorphism is an isomorphism of topological spaces.

• A diffeomorphism is an isomorphism of spaces equipped with a differential structure (differentiable manifolds).

• In geometry, isomorphisms and automorphisms are often called transformations (rigid, affine, projective transformations).

Category theory, which can be viewed as a formalization of the concept of mapping between structures, provides a language that may be used to unify the approach to these different aspects of the basic idea.