Comment on page


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:
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.)