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?
EDIT: In light of the 'uncear' close vote, I will attempt to be more specific.
Suppose we have a class $X$ with some binary operations $\{\odot_i\}_{i<n}\subseteq ^{X^2}X$$\{\odot_i\}_{i<n}\subseteq {}^{X^2}X$ and a collection of subsets $\{U_i\}_{i<\lambda}\subseteq\mathcal{P}(X)$. This may be, for example, a topological group or a valued module.
A-priori we do not necessarily know anything about the structure $S=(X,\{\odot_i\}_{i<n},\{U_i\}_{i<\lambda})$ until we begin writing down some conjectures $\{\phi_i\}_{i<m}$ and seeing if $S\vDash\phi_i$. For nontrivial $\phi_i$, one way to go about this is to begin proving simple or obvious sentences about $S$ and seeing if $\phi_i$ or $\neg\phi_i$ is a consequence of any combination of these simple sentences. If it isn't, we can gradually proceed to prove more and more complex sentences true until we reach a sufficient depth to begin tackling $\phi_i$. A great example here is number theory.
One alternative is to identify $S$ as a substructure of some structure $S'$ that we're very familiar with, then begin using well-known facts about $S'$ and its substructures to tackle $\phi_i$. This is what is meant by a 'top down' approach.
Another alternative route is to identify $S$ as containing a copy of another structure $S''$ we are very familiar with, then view $S$ as an extension of $S''$ in an appropriate sense to begin approaching $\phi_i$ from well known or deep facts abut $S''$. This is what is meant by a 'bottoms up' approach.
