Viewing parts of $\mathbb{V}$ 'from the top down' or 'from the bottom up'

Question

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 $\{\odot_i\}_{i<n}\subseteq {}^{X^2}X$ 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.

Answer 1

This is a central idea in many large cardinal axioms, which postulate the existence of a nontrivial elementary embedding of the set-theoretic universe $V$ into a transitive class $M$. $$j:V\to M$$ This situation conforms with your description because we may view the relationship between the universe $V$ and the class $M$ in two ways:

• The universe $V$ is smaller than $M$ because it is isomorphic to its image $j"V$, which is a subclass of $M$.
• The universe $V$ is larger than $M$ because $M\subset V$ is a transitive class in $V$.

Many arguments in large cardinal set theory involve at their essence the interaction of these two perspectives.

When there is such an embedding, then any sufficiently large mathematical structure $X$ admits the same two perspectives, comparing $X$ with $j(X)$. On the one hand, we can understand $X$ by understanding its version $j(X)$ in the other universe $M$, with the the isomorphic copy $j"X$ sitting inside $j(X)$. So we essentially have two versions of $X$, a smaller one $j"X$ and a larger one $j(X)$, each of them analogous to $X$ in slightly different ways, and we can view either one as the genuine $X$ in order to play off on your two perspectives, looking at $j"X$ from $j(X)$ or looking at $j(X)$ from $j"X$.

Answer 2

This is not what you are looking for, but I can't help myself.

Say the universe is a (closed) manifold $M$. One way of studying the manifold "from the bottom up" is by looking at the sublevel sets of a function $f:M\rightarrow \mathbb{R}$, i.e. by studying the topology of the sets

$$ M_c=\{x\in M\,|\, f(x)\leq c\}. $$

Looking at $M$ from the top down would be by looking at superlevel sets

$$ M^c=\{x\in M\,|\, f(x)\geq c\}. $$

But these things contain the same information: A sublevel set for $f$ is a superlevel set of $-f$. This observation leads to Poincaré duality.

Maybe more in spirit: Any $n$ dimenional manifold is made up from copies of $\mathbb{R}^n$, but is also a submanifold of $\mathbb{R}^{2n}$.