Free group universal property

The uni­ver­sal prop­erty of the free group ba­si­cally tells us that “the defi­ni­tion of the free group doesn’t de­pend (up to iso­mor­phism) on the ex­act de­tails of the set \(X\) we picked; only on its car­di­nal­ity”, which is morally a very use­ful thing to know. You may skip down to the next sub­head­ing if you might be scared of cat­e­gory the­ory, but the prop­erty it­self doesn’t need cat­e­gory the­ory and is helpful.

The uni­ver­sal prop­erty is the tech­ni­cal cat­e­gory-the­o­retic fact that the free-group func­tor is left ad­joint to the for­get­ful func­tor, and it is not so im­me­di­ately use­ful as the other more con­crete prop­er­ties on this page, but it is ex­ceed­ingly im­por­tant in cat­e­gory the­ory as a very nat­u­ral ex­am­ple of a pair of ad­joint func­tors and as an ex­am­ple for the gen­eral ad­joint func­tor the­o­rem.

State­ment and explanation

The uni­ver­sal prop­erty which char­ac­ter­ises the free group is:

The free group \(FX\) on the set \(X\) is the group, unique up to iso­mor­phism, such that for any group \(G\) and any func­tion of sets \(f: X \to G\) noteHere we’re slightly abus­ing no­ta­tion: we’ve writ­ten \(G\) for the un­der­ly­ing set of the group \(G\) here., there is a unique group ho­mo­mor­phism \(\overline{f}: FX \to G\) such that \(\overline{f}(\rho_{a_1} \rho_{a_2} \dots \rho_{a_n}) = f(a_1) \cdot f(a_2) \cdot \dots \cdot f(a_n)\).

This looks very opaque at first sight, but what it says is that \(FX\) is the unique group such that:

Given any tar­get group \(G\), we can ex­tend any map \(f: X \to G\) to a unique ho­mo­mor­phism \(FX \to G\), in the sense that when­ever we’re given the image of each gen­er­a­tor (that is, mem­ber of \(X\)) by \(f\), the laws of a group ho­mo­mor­phism force ex­actly where ev­ery other el­e­ment of \(FX\) must go. That is, we can spec­ify ho­mo­mor­phisms from \(FX\) by spec­i­fy­ing where the gen­er­a­tors go, and more­over, ev­ery pos­si­ble such speci­fi­ca­tion does in­deed cor­re­spond to a ho­mo­mor­phism.

Why is this a non-triv­ial prop­erty?

Con­sider the cyclic group \(C_3\) with three el­e­ments; say \(\{ e, a, b\}\) with \(e\) the iden­tity and \(a + a = b\), \(a+b = e = b+a\), and \(b+b = a\). Then this group is gen­er­ated by the el­e­ment \(a\), be­cause \(a=a\), \(a+a = b\), and \(a+a+a = e\). Let us pick \(G = (\mathbb{Z}, +)\). We’ll try and define a map \(f: C_3 \to \mathbb{Z}\) by \(a \mapsto 1\).

If \(C_3\) had the uni­ver­sal prop­erty of the free group on \(\{ e, a, b\}\), then we would be able to find a ho­mo­mor­phism \(\overline{f}: C_3 \to \mathbb{Z}\), such that \(\overline{f}(a) = 1\) (that is, mimick­ing the ac­tion of the set-func­tion \(f\)). But in fact, no such ho­mo­mor­phism can ex­ist, be­cause if \(\overline{f}\) were such a ho­mo­mor­phism, then \(\overline{f}(e) = \overline{f}(a+a+a) = 1+1+1 = 3\) so \(\overline{f}(e) = 3\), which con­tra­dicts that the image of the iden­tity un­der a group ho­mo­mor­phism is the iden­tity.

In essence, \(C_3\) “has ex­tra re­la­tions” (namely that \(a+a+a = e\)) which the free group doesn’t have, and which can thwart the at­tempt to define \(\overline{f}\); this is re­flected in the fact that \(C_3\) fails to have the uni­ver­sal prop­erty.

A proof of the uni­ver­sal prop­erty may be found el­se­where.


  • Free group

    The free group is “the purest way to make a group con­tain­ing a given set”.