Morphism

In mathematics, a morphism is an important idea from category theory. It generalizes many kinds of structure-preserving connections, like homomorphisms between algebraic structures, simple functions from one set to another, and continuous functions between topological spaces. While many examples of morphisms are maps, they don’t have to be. They can still be combined in a way that resembles how functions are put together.

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 consists of two types of things: objects and morphisms. Every morphism connects two objects, called its source and target. Think of a morphism like a path from one object to another.

In many common categories, objects are sets, and morphisms are functions that map one set to another. These morphisms can be combined in a specific way, called composition. This means you can link several morphisms together when 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, and associativity means the way you group the linked morphisms doesn’t change the overall 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 where if you follow it with two different morphisms and get the same result, those two morphisms must actually be the same. Think of it like a one-way path that doesn’t let different paths merge together.

Another type is an epimorphism, which is like the opposite of a monomorphism. For an epimorphism, if two morphisms followed by it give the same result, then the two morphisms must be the same.

When a morphism has an inverse — meaning you can “undo” it — it is called an isomorphism. This special kind of morphism shows that two objects are essentially the same in the context of the category, like two different shapes that can be perfectly matched up with each other.

Examples

In algebra, morphisms are often called homomorphisms. For example, between groups, rings, or modules, they preserve the structure. In topology, morphisms are continuous functions between spaces, and 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.