monomorphism

In category theory, a monomorphism, or mono, is a morphism f: A -> B that is left cancellable: for all objects C and all morphisms g1, g2: C -> A, f ∘ g1 = f ∘ g2 implies g1 = g2. Equivalently, f is determined by its precomposition with any map into A.

In the category of sets, monomorphisms are exactly the injective functions. In many algebraic categories such as groups, modules, rings, and abelian categories, monomorphisms correspond to injective homomorphisms. However, in more general categories a mono need not be an injective function on underlying elements, since such elements may not exist or be meaningful in the same way.

Examples include the inclusion of a subgroup H into a group G, or the inclusion of a subspace X into a topological space Y, both of which are monomorphisms. By contrast, a morphism that collapses information, such as a constant map, is generally not a mono.

Several equivalent characterizations exist: f is mono if and only if, for the pullback of f with itself, the resulting projections are equal, or equivalently the kernel pair of f is trivial. In many categories, monos represent subobjects of B; two monos with the same codomain define isomorphic subobjects.

See also: epimorphism, subobject, kernel, equalizer.

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