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Concept
Monomorphism
Summary
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation .
In the more general setting of , a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow f : X → Y such that for all objects Z and all morphisms g1, g2: Z → X,
Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below.
In the setting of posets intersections are idempotent: the intersection of anything with itself is itself. Monomorphisms generalize this property to arbitrary categories. A morphism is a monomorphism if it is idempotent with respect to .
The categorical dual of a monomorphism is an epimorphism, that is, a monomorphism in a category C is an epimorphism in the Cop. Every is a monomorphism, and every is an epimorphism.
Left-invertible morphisms are necessarily monic: if l is a left inverse for f (meaning l is a morphism and ), then f is monic, as
A left-invertible morphism is called a or a section.
However, a monomorphism need not be left-invertible. For example, in the category Group of all groups and group homomorphisms among them, if H is a subgroup of G then the inclusion f : H → G is always a monomorphism; but f has a left inverse in the category if and only if H has a normal complement in G.
A morphism f : X → Y is monic if and only if the induced map f∗ : Hom(Z, X) → Hom(Z, Y), defined by f∗(h) = f ∘ h for all morphisms h : Z → X, is injective for all objects Z.
Every morphism in a whose underlying function is injective is a monomorphism; in other words, if morphisms are actually functions between sets, then any morphism which is a one-to-one function will necessarily be a monomorphism in the categorical sense. In the the converse also holds, so the monomorphisms are exactly the injective morphisms.
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