Monomorphisms

Monomorphisms are a fundamental notion in category theory. A morphism f: A → B is called a monomorphism (or mono) if it is left-cancellable: for every object Z and every pair of morphisms g1, g2: Z → A, f ∘ g1 = f ∘ g2 implies g1 = g2. Equivalently, f is mono if it preserves distinctness when precomposed with any map into its domain.

In familiar categories, monos can be characterized more concretely. In the category Set of sets and functions, a function f: A → B is mono precisely when it is injective. In the category of groups (Grp), monomorphisms are exactly the injective group homomorphisms; in the category of topological spaces (Top), monomorphisms are exactly the injective continuous maps; in rings (Ring), monomorphisms are the injective ring homomorphisms.

Examples and non-examples help illustrate the concept. The inclusion map i: S ↪ A of a subset S into A is a mono. Projections or constant maps are typically not monos; for instance, the constant map from a nontrivial set to a singleton is not mono in Set.

Key properties include:

- Monomorphisms are closed under composition: if f: A → B and g: B → C are monos, then g ∘ f: A → C is a mono.

- Isomorphisms are monomorphisms, and every mono is a factor of an isomorphism in many contexts.

- Monomorphisms are stable under pullback; in particular, subobjects of an object correspond to equivalence classes of monos.

The dual notion is an epimorphism (epi), defined by right-cancellability. In many categories, monos and epis interact to characterize subobjects and quotient objects, with kernels often yielding monos in abelian settings.