Product (Category Theory)

This si­mul­ta­neously cap­tures the con­cept of a product of sets, posets, groups, topolog­i­cal spaces etc. In ad­di­tion, like any uni­ver­sal con­struc­tion, this char­ac­ter­i­za­tion does not differ­en­ti­ate be­tween iso­mor­phic ver­sions of the product, thus al­low­ing one to ab­stract away from an ar­bi­trary, spe­cific con­struc­tion.

Definition

Given a pair of ob­jects \(X\) and \(Y\) in a cat­e­gory \(\mathbb{C}\), the product of \(X\) and \(Y\) is an ob­ject \(P\) along with a pair of mor­phisms \(f: P \rightarrow X\) and \(g: P \rightarrow Y\) satis­fy­ing the fol­low­ing uni­ver­sal con­di­tion:

Given any other ob­ject \(W\) and mor­phisms \(u: W \rightarrow X\) and \(v:W \rightarrow Y\) there is a unique mor­phism \(h: W \rightarrow P\) such that \(fh = u\) and \(gh = v\).

Children:

  • Universal property of the product

    The product can be defined in a very gen­eral way, ap­pli­ca­ble to the nat­u­ral num­bers, to sets, to alge­braic struc­tures, and so on.

Parents:

  • Category theory

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