Morphism

In mathematics, a morphism is an important idea from category theory. It generalizes many kinds of connections that keep structure, like homomorphisms between algebraic structures, simple functions from one set to another, and continuous functions between topological spaces.

Morphisms connect two things called objects: the source and the target. There is a special way to combine, or “compose,” morphisms when the target of the first matches the source of the second. This composition follows rules similar to function composition, including associativity and the presence of an identity morphism for each object.

Morphisms and categories are used widely in modern mathematics. They were first introduced for areas like homological algebra and algebraic topology. They are also key tools in Grothendieck's scheme theory, which extends algebraic geometry to include algebraic number theory.

Definition

A category has two parts: objects and morphisms. Every morphism connects two objects. We call these objects its source and target. Think of a morphism as a path from one object to another.

In many categories, objects are sets. Morphisms are functions that map one set to another. These morphisms can be linked together in a special way, called composition. This means you can connect several morphisms if the target of one is the source of the next. Composition follows two important rules: identity and associativity. Identity means each object has a special morphism that acts like a "do nothing" path. Associativity means the way you group the linked morphisms doesn’t change the final result.

Some special morphisms

Morphisms can have special properties that make them important in mathematics.

One type is called a monomorphism. A monomorphism is a special kind of morphism. It is like a one-way path that doesn’t let different paths merge together.

Another type is an epimorphism. It is like the opposite of a monomorphism.

When a morphism has an inverse — meaning you can “undo” it — it is called an isomorphism. This shows that two objects are essentially the same in the category, like two different shapes that match up perfectly.

Examples

In algebra, morphisms are sometimes called homomorphisms. They help keep the structure the same between groups, rings, or modules. In topology, morphisms are continuous functions between spaces. Some special ones are called homeomorphisms.

Other examples include smooth functions between smooth manifolds, functors between small categories, and natural transformations in functor categories. For more information, see Category theory.