The definitions of "natural transformation", "natural isomorphism between functors", and "natural isomorphism between objects" captures – among other things – the intuitive notion of "an isomorphism that does not depend on an arbitrary choice" (from Wikipedia). The standard example is a finite dimensional vector space $V$ being naturally isomorphic to its double dual $V$** because the isomorphism doesn't depend on the choice of basis.
What I wonder:
Does this informal notion of choice have to do
with the formal notion of choice in the axiom of
choice?Is "being naturally
isomorphic" somehow related
to "being provably isomorphic from $ZF$ without $AC$"?Or are these completely unrelated concepts?
Best Answer
"Naturality" in a categorical sense is much more than "not depending on choices", and, also, is essentially unrelated to issues about the Axiom of Choice.
In the example of vector spaces over a field, we can look at the category of _finite_dimensional_ vector spaces, to avoid worrying about using AxCh to find elements of the dual. The non naturality of any isomorphisms of finite-dimensional vectorspaces with their duals resides in the fact that, provably, as a not-hard exercise, there is no collection of isomorphisms $\phi_V:V\rightarrow V^*$ of isomorphisms of f.d. v.s.'s $V$ to their duals, compatible with all v.s. homs $f:V\rightarrow W$.
In contrast, the isomorphism $\phi_V:V\rightarrow V^{**}$ to the second dual, by $\phi_V(v)(\lambda)=\lambda(v)$ is compatible with all homs, as an easy exericise! This latter compatibility is the serious meaning of "naturality".
True, if capricious or random choices play a role, the chance that the outcome is natural in this sense is certainly diminished! But that aspect is not the defining property!
Edit (16 Apr '12): as alancalvitti notes, the ubiquity of adjunctions, and the naturality and sense of "naturality", and counter-examples to naive portrayals, deserve wider treatment at introductory levels. After all, this can be done with almost no serious "formal" category-theoretic overhead, and pays wonderful returns, at the very least organizing one's thinking. Distinguishing "characterization" from "construction-to-prove-existence" is related. E.g., "Why is the product topology so coarse?": to say that "it's the definition" is unhelpful; to take the categorical definition of "product" and _find_out_ what topology on the cartesian product of sets is the categorical product topology is a do-able, interesting exercise! :)