Given a set $S$, here are 5 ways of thinking about elements of $S$, in increasing abstraction:
- an actual element, e.g. $s\in S$
- an inclusion map, e.g. $i_s:\{s\}\hookrightarrow S$
- an equivalence class of maps from singletons, e.g. $\{f:X\rightarrow S\mid X\text{ is a singleton}\}/\sim$, where $(f:X\rightarrow S)\sim (g:Y\rightarrow S)$ if there is a bijection $j:X\rightarrow Y$ such that $f=g\circ j$
- a map from a singleton $f:{\ast}\rightarrow S$
- a map $f:T\rightarrow S$, where $T$ is any set
While none of these definitions are particularly complicated, I think the process of reaching definitions 4 and 5 from definition 1 is one of the most important achievements of modern mathematics. What I see as two of the key aspects: the shift of focus from elements to maps, and the shift from trying to take equivalence classes of isomorphic things to "not caring" (i.e. taking any one of them). The same conceptual leaps can be made almost anywhere - the kernel of a group homomorphism $f:G\rightarrow H$, instead of being a subset $\ker(f)\subseteq G$, can be defined to be a map $k:K\rightarrow G$ satisfying a universal property (which, notably, only specifies $K$ and $k$ up to (unique) isomorphism).
But however profoundly imaginative and illuminating I find the above way of thinking, if someone has not seen situations in their mathematics where this kind of approach is useful, or at least clarifying, they may very well consider it to be unnecessary abstraction. Here is one particular case that I was arguing about recently: submanifolds of a manifold $M$. I posited that a much more aesthetic definition of immersed submanifold would simply be "an immersion $f:N\rightarrow M$" (possibly required to be injective), and an embedded submanifold would simply be "a smooth embedding $f:N\rightarrow M$". But I don't know enough about the theory of manifolds to give examples of where such an approach is helpful, in either a practical sense (it helps us prove theorems) or just in providing intuition, or even to know for sure if it's even really a good alternative definition.
So, I would like to ask for examples of where the kind of abstractions I'm talking about have advanced some aspect of mathematics - better theorems would be ideal, but better intuition is good too - and the more accessible the better. The huge example I know of is the relative point of view in algebraic geometry, but this would be hard to explain to someone who wasn't already familiar with schemes and their morphisms. I would be hoping for examples that require (at least mildly) less machinery. Also, I am particularly interested in whether my submanifold definitions are useful, and if so, whether there are any resources (either in a book or online) for differential topology where this kind of abstract approach is emphasized.