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

1. A general morphism is called a homomorphism,

2. A morphism
in a category is a monomorphism
if, for any two morphisms ,
implies that ,

3. A morphism
in a category is an epimorphism
if, for any two morphisms ,
implies ,

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. https://mathworld.wolfram.com/Morphism.html

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