nLab morphism

Idea

The notion of morphism in category theory is an abstraction of the notion of homomorphism.

In a general category, a morphism is an arrow between two objects. The word map is also sometimes used as a synonym.

Definition

Given two objects in a (locally small) category, say $x$ and $y$, there is a set $hom(x,y)$, called a hom-set, whose elements are morphisms from $x$ to $y$. Given a morphism $f$ in this hom-set, we write $f:x \to y$ to indicate that it goes from $x$ to $y$.

More generally, a morphism is what goes between objects in any n-category.

Examples

The most familiar example is the category Set, where the objects are sets and the morphisms are functions. Here if $x$ and $y$ are sets, a morphism $f: x \to y$ is a function from $x$ to $y$.

Related concepts

• object

• morphism, multimorphism

  • inverse morphism

  • adjoint morphism

  • heteromorphism

  • endomorphism, automorphism, monomorphism, epimorphism

• 1-morphism

  horizontal morphism, vertical morphism

• 2-morphism

• 3-morphism

• k-morphism