Concepts

Homomorphism (Algebra) 同态

• Wikipedia: homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).

• Roughly speaking, homomorphism = mapping (one direction, not necessarily surjective) + preserving mathematical structures.

• Of the same type:

  • unit quaternions are isomorphic to SU(2) -> not rigorous, because quaternion by itself is a number system, not a group

  • the group of unit quaternions (with multiplication as the binary operation, implicitly) is isomorphic to SU(2) -> good

Homeomorphism (Topology) 同胚

• Not to be confused with homomorphism in algebra, which does not imply bijection.

• Homeomorphism is a concept in topology, which implies bijection + preserving mathematical structures. It is an isomorphism of topological spaces.

Isomorphism 同构

• Wikipedia: Isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping.

• Roughly speaking, isomorphism = bijection + preserving mathematical structures.

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

• 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, typically differentiable manifolds.

• A permutation is an automorphism of a set.

• In geometry, isomorphisms and automorphisms are often called transformations, for example rigid transformations, affine transformations, projective transformations.

More concepts and topics to discuss:

• Charts, Atlas, Connected, Simply connected, Projective space

• The symbol ≅ can in principle be used to designate an isomorphism in any category (e.g., isometric, diffeomorphic, homeomorphic, linearly isomorphic, etc.)