# 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](https://en.wikipedia.org/wiki/Isometry) is an isomorphism of [metric spaces](https://en.wikipedia.org/wiki/Metric_space).
* **A** [**homeomorphism**](https://en.wikipedia.org/wiki/Homeomorphism) **is an isomorphism of** [**topological spaces**](https://en.wikipedia.org/wiki/Topological_space)**.**
* A [diffeomorphism](https://en.wikipedia.org/wiki/Diffeomorphism) is an isomorphism of spaces equipped with a [differential structure](https://en.wikipedia.org/wiki/Differential_structure), typically [differentiable manifolds](https://en.wikipedia.org/wiki/Differentiable_manifold).
* A [permutation](https://en.wikipedia.org/wiki/Permutation) is an automorphism of a [set](https://en.wikipedia.org/wiki/Set_\(mathematics\)).
* In [geometry](https://en.wikipedia.org/wiki/Geometry), isomorphisms and automorphisms are often called [transformations](https://en.wikipedia.org/wiki/Transformation_\(function\)), for example [rigid transformations](https://en.wikipedia.org/wiki/Rigid_transformation), [affine transformations](https://en.wikipedia.org/wiki/Affine_transformation), [projective transformations](https://en.wikipedia.org/wiki/Projective_transformation).

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.)


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