Universal property

In cat­e­gory the­ory, we at­tempt to avoid think­ing about what an ob­ject is, and look only at how it in­ter­acts with other ob­jects. It turns out that even if we’re not al­lowed to talk about the “in­ter­nal” struc­ture of an ob­ject, we can still pin down some ob­jects just by talk­ing about their in­ter­ac­tions. For ex­am­ple, if we are not al­lowed to define the empty set as “the set with no el­e­ments”, we can still define it by means of a “uni­ver­sal prop­erty”, talk­ing in­stead about the func­tions from the empty set rather than about the el­e­ments of the empty set.

This page is not de­signed to teach you any par­tic­u­lar uni­ver­sal prop­er­ties, but rather to con­vey a sense of what the idea of “uni­ver­sal prop­erty” is about. You are sup­posed to let it wash over you gen­tly, with­out wor­ry­ing par­tic­u­larly if you don’t un­der­stand words or even en­tire sen­tences.

Examples

  • The empty set can be defined by a uni­ver­sal prop­erty. Speci­fi­cally, it is an in­stance of the idea of an ini­tial ob­ject, in the cat­e­gory of sets. The same idea cap­tures the triv­ial group, the ring \(\mathbb{Z}\) of in­te­gers, and the nat­u­ral num­ber \(0\).

  • The product has a uni­ver­sal prop­erty, gen­er­al­is­ing the set product, the product of in­te­gers, the great­est lower bound in a par­tially or­dered set, the prod­ucts of many differ­ent alge­braic struc­tures, and many other things be­sides.

  • The free group has a uni­ver­sal prop­erty (we re­fer to this prop­erty by the un­wieldy phrase “the free-group func­tor is left-ad­joint to the for­get­ful func­tor”). The same prop­erty can be used to cre­ate free rings, the dis­crete topol­ogy on a set, and the free semi­group on a set. This idea of the left ad­joint can also be used to define ini­tial ob­jects (which is the gen­er­al­ised ver­sion of the uni­ver­sal prop­erty of the empty set). noteIn­deed, an ini­tial ob­ject of cat­e­gory \(\mathcal{C}\) is ex­actly a left ad­joint to the unique func­tor from \(\mathcal{C}\) to \(\mathbf{1}\) the one-ar­row cat­e­gory.

The above ex­am­ples show that the ideas of cat­e­gory the­ory are very gen­eral. For in­stance, the third ex­am­ple cap­tures the idea of a “free” ob­ject, which turns up all over ab­stract alge­bra.

Defi­ni­tion “up to iso­mor­phism”

ex­plain that we only usu­ally get things defined up to iso­mor­phism, and what that means anyway

Univer­sal prop­er­ties might not define objects

Univer­sal prop­er­ties are of­ten good ways to define things, but just like with any defi­ni­tion, we always need to check in each in­di­vi­d­ual case that we’ve ac­tu­ally defined some­thing co­her­ent. There is no silver bul­let for this: uni­ver­sal prop­er­ties don’t just mag­i­cally work all the time.

For ex­am­ple, con­sider a very similar uni­ver­sal prop­erty to that of the empty set (de­tailed here), but in­stead of work­ing with sets, we’ll work with fields, and in­stead of func­tions be­tween sets, we’ll work with field ho­mo­mor­phisms.

The cor­re­spond­ing uni­ver­sal prop­erty will turn out not to be co­her­ent:

The ini­tial field noteA­nalo­gously with the empty set, but fields can’t be empty so we’ll call it “ini­tial” for rea­sons which aren’t im­por­tant right now. is the unique field \(F\) such that for ev­ery field \(A\), there is a unique field ho­mo­mor­phism from \(F\) to \(A\).

(The slick way to com­mu­ni­cate this proof to a prac­tised math­e­mat­i­cian is “there are no field ho­mo­mor­phisms be­tween fields of differ­ent char­ac­ter­is­tic”.)

It will turn out that all we need is that there are two fields \(\mathbb{Q}\) and \(F_2\) the field on two el­e­ments. %%note: \(F_2\) has el­e­ments \(0\) and \(1\), and the re­la­tion \(1 + 1 = 0\).%%

Sup­pose we had an ini­tial field \(F\) with mul­ti­plica­tive iden­tity el­e­ment \(1_F\); then there would have to be a field ho­mo­mor­phism \(f\) from \(F\) to \(F_2\). Re­mem­ber, \(f\) can be viewed as (among other things) a group ho­mo­mor­phism from the mul­ti­plica­tive group \(F^*\) %%note: That is, the group whose un­der­ly­ing set is \(F\) with­out \(0\), with the group op­er­a­tion be­ing “mul­ti­pli­ca­tion in \(F\)”.%% to \(F_2^*\).

Now \(f(1_F) = 1_{F_2}\) be­cause the image of the iden­tity is the iden­tity, and so \(f(1_F + 1_F) = 1_{F_2} + 1_{F_2} = 0_{F_2}\).

But field ho­mo­mor­phisms are ei­ther in­jec­tive or map ev­ery­thing to \(0\) (proof); and we’ve already seen that \(f(1_F)\) is not \(0_{F_2}\). So \(f\) must be in­jec­tive; and hence \(1_F + 1_F\) must be \(0_F\) be­cause \(f(1_F + 1_F) = 0_{F_2} = f(0_F)\).

Now ex­am­ine \(\mathbb{Q}\). There is a field ho­mo­mor­phism \(g\) from \(F\) to \(\mathbb{Q}\). We have \(g(1_F + 1_F) = g(1_F) + g(1_F) = 1 + 1 = 2\); but also \(g(1_F + 1_F) = g(0_F) = 0\). This is a con­tra­dic­tion. <div><div>

Children:

Parents:

  • Category theory

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