# Monomorphism

*in a category*

A morphism $\mu : A \to B$ of a category $\mathfrak{K}$ for which $\alpha \, \mu = \beta \, \mu$ ($\alpha$, $\beta$ from $\mathfrak{K}$) implies that $\alpha = \beta$ (in other words, $\mu$ can be cancelled on the right). An equivalent definition of a monomorphism is: For any object $X$ of a category $\mathfrak{K}$ the mapping of sets induced by $\mu$, $$ \operatorname{Hom} \left({X, A}\right) \to \operatorname{Hom} \left({X, B}\right), $$

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 $\mathfrak{K}$ forms a subcategory of $\mathfrak{K}$ (usually denoted by $\operatorname{Mon} \mathfrak{K}$).

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, $\alpha \, \mu$ 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.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Monomorphism&oldid=40171