# 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)