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.