I am curious about instances where we can glean nontrivial information about a certain piece of the universe by viewing it as being 'built over' a smaller part of the universe, or alternatively viewing it as a subspace of a larger piece of the universe.

For 'top down' examples we can view any ordered field as a subfield of the Surreals, or any Archimedean ordered field as a subfield of the Reals, or any linearly ordered group as an ordered additive subgroup of $\mathbb{R}^\Gamma$ for $\Gamma$ the collection of Archimedean equivalence classes of the group.

For 'bottom up' examples we can view all abelian groups as $\mathbb{Z}$-modules, or any ordered field as an extension of $\mathbb{Q}$ -- category theory also probably has some nice examples, but it is not my forte.

What are other cool/useful examples of this occurring?