What is a natural isomorphism?

Solution 1:

"Natural" refers to something coming from a natural transformation between two functors (functors being maps between categories). In particular, a natural transformation is a natural isomorphism when each of its components are isomorphisms.

As explained in the Wikipedia article, let $C$ and $D$ be categories (which might be the same), and let $F:C\rightarrow D$ and $G:C\rightarrow D$ be functors (which also might be the same). Then a natural transformation is just a collection of morphisms $\eta_X:F(X)\rightarrow G(X)$ in $D$, one for each $X\in\text{Ob}(C)$, such that for any morphism $f:X\rightarrow Y$ in $C$ we have $$\eta_Y\circ F(f)=G(f)\circ \eta_X$$

Intuitively, the idea of a natural transformation is that it is a map between functors, just like a functor is a map between categories. This can be made rigorous by constructing a new category (called the functor category $\text{Funct}(C,D)$), whose objects are functors from $C$ to $D$, and the morphisms are precisely the natural transformations.

The examples provided in the Wikipedia article are also good illustrations of how natural transformations occur - if you need help understanding them, perhaps you can ask a question specifically about the example.

Solution 2:

In complement to Zev's answer, the natural adjective is used in natural transformation for several converging reasons, linked to the roots of category theory in homology and representation theory of algebraic objects.

What is natural is that a set of functors between the same two categories ($C$ and $D$ to take Zev's notation) can be seen as a category if its arrows (transformation between functors) preserves key topological properties of the relation between $C$ and $D$.

To transfer that to a limited but more explicit family of examples, if you see the functors as calculating invariants, characteristic classes, derived objects, etc. of objects in $C$ (algebraic objects, spaces, functions, etc.) as members of $D$ (groups, rings, set cardinals, trees, etc.), the commutativity of the diagram between functors and a natural transformation produces very strong constraints on both and allows you to select mathematical notions (i.e. specific functors) according to their mutual compatibility and consistency, their network of relationships (natural transformations) becoming the real objects of study (the natural ones some mathematicians would say) and subsuming the initial study of the objects in the category $C$ through the invariants expressed in $D$, if $D$ is well chosen (see also the Yoneda Lemma when $C=D$).

This is one way to see the path from the study of polynomial equations to the study of Galois theory, or the path from the study of symmetry of regular geometrical objects to the study of representations of groups among others (see one of the other answers for illustration with homotopy groups).

This gives you a hint that category theory is a powerful accelerator of abstraction and of exploration of regular concepts. Naturalness has also its drawbacks but this is not the subject here.

Solution 3:

Naturality at a lower level sometimes refers to functoriality: for example, the fundamental group $\pi_1$ is "natural" because $\pi_1$ is a functor $\mathbf{Top}^* \to \mathbf{Grp}$, so if you have a continuous map $f : X \to Y$, then you get a homomorphism $f_* : \pi_1(X, x) \to \pi_1(Y, f(x))$.

An example of a "natural" isomorphism is the canonical isomorphism between a finite-dimensional vector space $V$ and its double-dual $V^{**}$. Again, this is natural because the double-dual $(-)^{**}$ is a functor $\mathbf{Vec} \to \mathbf{Vec}$. It's canonical because it's choice-free, in some sense. It's also natural in the technical sense: there is a natural transformation $\eta$ from the identity functor to the double-dual functor $(-)^{**}$, and the component $\eta_V : V \to V^{**}$ of $\eta$ at each finite-dimensional vector space is an isomorphism.