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# Concepts

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

- 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.

- 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:

- A diffeomorphism is an isomorphism of spaces equipped with a differential structure, typically differentiable manifolds.
- 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.)

Last modified 6mo ago