# Morphism A **morphism** is a structure-preserving [map](https://en.wikipedia.org/wiki/Map_(mathematics) "Map (mathematics)") from one [mathematical structure](https://en.wikipedia.org/wiki/Mathematical_structure) to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In [set theory](https://en.wikipedia.org/wiki/Set_theory), morphisms are [functions](https://en.wikipedia.org/wiki/Function_(set_theory)); in [linear algebra](https://en.wikipedia.org/wiki/Linear_algebra "Linear algebra"), [linear transformations](https://en.wikipedia.org/wiki/Linear_transformations "Linear transformations"); in [group theory](https://en.wikipedia.org/wiki/Group_theory "Group theory"), [group homomorphisms](https://en.wikipedia.org/wiki/Group_homomorphism "Group homomorphism"); in [topology](https://en.wikipedia.org/wiki/Topology "Mathematical structure"), [continuous functions](https://en.wikipedia.org/wiki/Continuous_functions "Continuous functions"), and so on. ### Endomorphism An **endomorphism** is a [morphism](https://en.wikipedia.org/wiki/Morphism) from a [mathematical object](https://en.wikipedia.org/wiki/Mathematical_object) to itself. An endomorphism that is also an [isomorphism](https://en.wikipedia.org/wiki/Isomorphism "Isomorphism") is an [automorphism](https://en.wikipedia.org/wiki/Automorphism "Automorphism"). For example, an endomorphism of a [vector space](https://en.wikipedia.org/wiki/Vector_space "Vector space") _V_ is a [linear map](https://en.wikipedia.org/wiki/Linear_map "Morphism") _f_: _V_ → _V_ --- Date: 20210518 Links to: [Mathematics MOC](Mathematics%20MOC.md) References: * [Wikipedia](https://en.wikipedia.org/wiki/Morphism) * [Wikepedia, endomorphism](https://en.wikipedia.org/wiki/Endomorphism)