# Monomorphism

*in a category*

A morphism of a category for which ( from ) implies that (in other words, can be cancelled on the right). An equivalent definition of a monomorphism is: For any object of a category the mapping of sets induced by ,

must be injective. The product of two monomorphisms is a monomorphism. Each left divisor of a monomorphism is a monomorphism. The class of all objects and all monomorphisms of an arbitrary category forms a subcategory of (usually denoted by ).

In the category of sets (cf. Sets, category of) monomorphisms are the injections (cf. Injection). Dual to the notion of a monomorphism is that of an epimorphism.

#### References

[1] | M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian) |

[2] | I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968) |

#### Comments

In the first definition above, composition of morphisms is written in "diagram order" (that is, means "a followed by m" ). If, as is frequently done, the opposite convention is employed, then monomorphisms are morphisms which can be cancelled on the left.

#### References

[a1] | S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7 |

**How to Cite This Entry:**

Monomorphism. O.A. Ivanova (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Monomorphism&oldid=15107