A morphism is a map between two objects in an abstract category.

1. A general morphism is called a homomorphism,

2. A morphism f:Y->X in a category is a monomorphism if, for any two morphisms u,v:Z->Y, fu=fv implies that u=v,

3. A morphism f:Y->X in a category is an epimorphism if, for any two morphisms u,v:X->Z, uf=vf implies u=v,

4. A bijective morphism is called an isomorphism (if there is an isomorphism between two objects, then we say they are isomorphic),

5. A surjective morphism from an object to itself is called an endomorphism, and

6. An isomorphism between an object and itself is called an automorphism.

See also

Automorphism, Category, Category Theory, Epimorphism, Homeomorphism, Homomorphism, Isomorphism, Monomorphism, Object, Separable Morphism

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Cite this as:

Weisstein, Eric W. "Morphism." From MathWorld--A Wolfram Web Resource.

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