Category (mathematics)

A category consists of a collection of objects with morphisms between them. A morphism \(f\) goes from one object, say \(X\), to another, say \(Y\), and is drawn as an arrow from \(X\) to \(Y\). Note that \(X\) may equal \(Y\) (in which case \(f\) is referred to as an endomorphism). The object \(X\) is called the source or domain of \(f\) and \(Y\) is called the target or codomain of \(f\), though note that \(f\) itself need not be a function and \(X\) and \(Y\) need not be sets. This is written as \(f: X \rightarrow Y\).

These morphisms must satisfy three conditions:

  1. Composition: For any two morphisms \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z\), there exists a morphism \(X \rightarrow Z\), written as \(g \circ f\) or simply \(gf\).

  2. Associativity: For any morphisms \(f: X \rightarrow Y\), \(g: Y \rightarrow Z\) and \(h:Z \rightarrow W\) composition is associative, i.e., \(h(gf) = (hg)f\).

  3. Identity: For any object \(X\), there is a (unique) morphism, \(1_X : X \rightarrow X\) which, when composed with another morphism, leaves it unchanged. I.e., given \(f:W \rightarrow X\) and \(g:X \rightarrow Y\) it holds that: \(1_X f = f\) and \(g 1_X = g\).


  • Category theory

    How mathematical objects are related to others in the same category.