Monomorphism

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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 X \hookrightarrow Y.

In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism, that is, an arrow f : XY such that, for all morphisms g1, g2 : ZX,

Monomorphisms are a categorical generalization of injective functions; in some categories the notions coincide, but monomorphisms are more general, as in the examples below.

The categorical dual of a monomorphism is an epimorphism, i.e. a monomorphism in a category C is an epimorphism in the dual category Cop. Every section is a monomorphism, and every retraction is an epimorphism.

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Terminology

The companion terms monomorphism and epimorphism were originally introduced by Nicolas Bourbaki; Bourbaki uses monomorphism as shorthand for an injective function. Early category theorists believed that the correct generalization of injectivity to the context of categories was the cancellation property given above. While this is not exactly true for monic maps, it is very close, so this has caused little trouble, unlike the case of epimorphisms. Saunders Mac Lane attempted to make a distinction between what he called monomorphisms, which were maps in a concrete category whose underlying maps of sets were injective, and monic maps, which are monomorphisms in the categorical sense of the word. This distinction never came into general use.

Another name for monomorphism is extension, although this has other uses too.

Relation to invertibility

Left invertible maps are necessarily monic: if l is a left inverse for f (meaning l \circ f = \operatorname{id}_{X}), then f is monic, as

A left invertible map is called a split mono.

A map f : XY is monic if and only if the induced map f : Hom(Z, X) → Hom(Z, Y), defined by f_{*}h = f \circ h for all morphisms h : ZX , is injective for all Z.

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