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Given a set $S$, here are 5 ways of thinking about elements of $S$, in increasing abstraction:

  1. an actual element, e.g. $s\in S$
  2. an inclusion map, e.g. $i_s:\{s\}\hookrightarrow S$
  3. 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 $f=g\circ j$
  4. a map from a singleton $f:{\ast}\rightarrow S$
  5. 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.

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up vote 6 down vote accepted

Ultimately, the abstraction in 5 above is well encapsulated in the Yoneda lemma. In a general category, there's no way to dig into the "internal structure" of an object. On the other hand, one can in practice for many concrete categories (sets, groups, topological spaces, ...) and even not-so-concrete ones (simplicial sets, opposite categories of concrete ones, ...) see the objects in a manner more individualized than the abstraction of category theory can afford. So why should we even bother with this arrow-theoretic approach?

One is that ultimately, proving things in an arrow-theoretic context is much more efficient than proving them for each specific category. And even if one wanted to prove them for each specific category, the cleanest approach seems often to be to use the universal property. Here is a simple example. Suppose one wanted to show that the abelianization of the free group on a set $S$ was the free abelian group on $S$. There are several approaches one could take:

  1. One could use the (somewhat ad hoc) construction of a free group as words on $S$, and the construction of the free abelian group as commutative words on $S$. One could define a map from the words on $S$ to commutative words on $S$ (i.e. $\mathbb{Z}[S]$). One could then check that the kernel of this map was the commutator subgroup...

  2. One could argue that mapping from the the abelianization of the free group $G_S$ on $S$ generators into an abelian group $A$ is the same thing as mapping from the free group itself, because of the universal property of the abelianization. To map from the free group on $S$ to $A$ is to give a map of sets $S \to A$. This is the same universal property as that of the free abelian group. By the Yoneda lemma, the two objects are isomorphic--since the descriptions of how to map out of them are identical.

  3. If one wished to be a bit more slick, one could easily consider the appropriate functors between the categories of abelian groups, the category of groups, and the category of sets (i.e. abelianization, taking free groups, forgetting structure) and then the statement is just a special case of the fact that taking the adjoint of a functor is (up to natural isomorphism) an anti-involution.

At least to me, argument 1 seems rather clunky. Arguments 2 and 3 seem much quicker and more convincing (at least once one buys the Yoneda lemma). To me, at least, it seems that viewing a free group via the description of how to map out of it---which by Yoneda is enough to describe it completely---is much more satisfying than the explicit construction. (Here's a similar exercise that might be even clunkier without arrows: show that the abelianization functor preserves pushouts. Viewed categorically this presents little difficulty (it's a left adjoint), but if one wished to reason strictly using the word construction, it might be less pleasant.)

In much of algebraic geometry, in fact, the above philosophy--describing an object by how one can map into (or out of) it predominates: projective space, Grassmannians, the Hilbert scheme and others can all be described (and often are most compactly described!) in this way. The last parenthetical comment actually means that one mightn't have a simple concrete description (e.g. by explicit polynomial equations cutting out a variety) of the object (as in the case of the Hilbert scheme). It follows that characterizing objects by how one can map into them can be very important for proving properties of these objects. As one example, the valuative criterion for properness lets you deduce that (the connected components of) the Hilbert scheme is proper.

Nonetheless, the above group-theoretic example is admittedly a rather straightforward statement, let me try to give an example where the categorical philosophy is the only way even to get a handle on the definitions. As you observed, the definition of a kernel (or cokernel) in an abelian category can only be defined via the inherently arrow-theoretic universal property. A standard example where you have to use this universal property (i.e., it's not otherwise obvious what to choose) is the category of sheaves on a topological space (or more generally on a site). In this case one wants to say that the cokernel is just what you would expect: you take the pointwise (i.e., open set by open set) quotient by the pointwise image. But this doesn't work in the sheaf category: the associated object is generally not a sheaf. (It turns out, fortunately, that something very close to what one would intuitively expect is true: one has to apply the functor of sheafification (which is the left adjoint of the inclusion of sheaves in the category of presheaves) to the "pointwise" cokernel, which is in general only a presheaf.)

So, if you care about sheaves (and ultimately, the general theory of sheaves has had a massive impact on modern mathematics, one example being the construction of various cohomology theories on varieties as flavors of sheaf cohomology, with one standard example being the proofs of the Weil conjectures), then you need this arrow-theoretic philosophy to make sense of the homological constructions.

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Akhil, did you mean: one has to apply sheafification (the left adjoint of the inclusion of the full subcategory of sheaves in the category of presheaves) to the pointwise cokernel? –  Rasmus May 13 '11 at 6:55
@Rasmus: Dear Rasmus, indeed. Thanks for the correction. –  Akhil Mathew May 13 '11 at 14:28
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Here's a simple question: why does the product of two groups have the same underlying set as the Cartesian product of sets, but the coproduct of two groups is not even close to the same underlying set as the disjoint union of sets? Well, the latter would be silly, but if you know enough category theory to know that the product and coproduct are dual to each other, you might wonder what underlying principle is responsible for breaking the duality.

The reason is that the forgetful functor $\text{Grp} \to \text{Set}$ is representable; it's $\text{Hom}(\mathbb{Z}, -)$. Covariant representable functors preserve limits, but not necessarily colimits. On the other hand, contravariant representable functors preserve colimits, but not necessarily limits. So that's what's breaking the duality, and now we know that it's valuable to think of elements of groups as morphisms from $\mathbb{Z}$.

Ultimately, though, if the goal is to make someone understand why the arrow-theoretic ideas are preferable to working directly with a particular concretization, why not introduce them to important categories that are not concretizable? Perhaps one of the most important examples is the homotopy category of topological spaces. If you really want to understand this category, you really have to get comfortable with using functors that are not faithful, such as $\text{Hom}(S^1, -)$ (the fundamental group, after pointing).

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