When and why do universal objects have extra properties?

Question

I'm interested in situations where universal objects come with more structure than their definitions suggest. A classic case of this is where the free abelian group on one element has a ring structure. Proving this is a straightforward exercise using the free-underlying adjunction. So I'd like to know of other cases of this phenomenon, and if possible an explanation as to why the extra structure comes about.

Plenty of examples are given in Hazewinkel's paper Niceness Theorems, but how about very familiar examples such as the rationals? Does, say, the characterisation of $\langle \mathbb{Q}, \gt \rangle$ as the Fraïssé limit of the category of finite linearly ordered sets and order preserving injections tell us why it should support a compatible group, ring and even field structure? Do characterisations of the reals relate to each other?

My question is not completely unrelated to Theorems for nothing (and the proofs for free), as shown by the example given there of subgroups of free groups being free, which also occurs in Hazewinkel's paper.

Answer 1

Always.

In the case of algebraic theories, the proof is simple. Let $F : Set \to V$ be the free functor, where $V$ is some algebraic theory. Then if $X$ is a set and $U$ a $V$-algebra,

$$ \operatorname{Hom}_{V}(F(X),U) \cong \operatorname{Hom}_{\operatorname{Set}}(X,|U|) $$

The right-hand side is a $V$-algebra, naturally in both $X$ and $U$. Therefore $F(X)$ is a co-$V$-algebra object in $V$, which is more structured than just being a $V$-algebra. Furthermore, if $X$ is a monoid then $F(X)$ becomes a Tall-Wraith $V$-monoid (obligatory n-lab reference: Tall-Wraith monoids).

(The details of this particular argument are a part of a paper by Sarah Whitehouse and myself on Tall-Wraith monoids.)

Answer 2

An incomplete answer on the subject of countable dense linear orders without endpoints;

I left some other thoughts at the Cafe; on further reflection, one can think of the maps $\cdot\times \frac{p}{q}:q[j,k]\rightarrow[pj,pk]$ defined on finite sets of integers as both inducing refinements of some map $[j,k]\rightarrow D$ and as rational scalings of the image.

(I don't know much about Fraïssé constructions, so this gets vague now) then from universality, we (almost) get a $\mathbb{Q}$-module structure on the Fraïssé limit?