Natural isomorphisms: what is the status now of "the Eilenberg/Mac Lane Thesis"?

The Church/Turing Thesis that we all know and love asserts that every algorithmically computable function (in an informally characterized sense) is in fact recursive/Turing computable/lambda computable. A certain intuitive concept, the Thesis claims, in fact picks out the same functions as certain (provably equivalent) sharply defined concepts.

Evidence? Two sorts: (1) "quasi-empirical", i.e. no unarguable clear exceptions have been found, (2) conceptual, as in for example Turing's own efforts to show that when we reflect on what we mean by algorithmic computation we get down to the sort of operations that a Turing machine can emulate.

OK, now compare. The "Eilenberg/Mac Lane Thesis" in one version (but does anyone call it that??) is that if an isomorphism between widgets and wombats is intuitively "natural" (i.e. doesn't depend on arbitrary choices of co-ordinates, or the like) then it can be regimented as an natural isomorphism between suitable functors in the official category-theoretic sense. A certain intuitive concept, the Thesis claims, in fact picks out the same isomorphisms as a certain sharply defined concept.

Evidence? We'd expect two sorts. (1*) "quasi-empirical", i.e. no clear exceptions. (2*) conceptual ...

Two main questions:

(A) But are there no exceptions? Or (to take the perhaps more likely direction of failure) are there well known cases where we can say "Hey, this is the sort of isomorphism whose intuitive naturalness was surely of the kind that Eilenberg/Mac Lane were trying to chraracterize, back in the day: but actually, you can't shoehorn this case into the framework of their theory of natural isomorphisms."

(B) Assuming the Thesis isn't defeated by counter-example, what are the best efforts at trying to show that, conceptually, it "ought" to be true?

Solution 1:

Does the following example count?

Let $X$ be a smooth quasi-projective scheme over some field. The Chern character $\mathrm{ch}_X : K_0(X) \to A(X,\mathbb{Q})$ is natural with respect to the contravariant functor structures (pullbacks). However, it is not natural with respect to the covariant functor structures (pushforwards). That is, for a smooth morphism $f : X \to Y$, we don't have $\mathrm{ch}_Y \circ f_{!} = f_* \circ \mathrm{ch}_X$ in general. However, the celebrated Grothendieck-Riemann-Roch theorem asserts that Todd classes may repair this: We have $\mathrm{td}_Y \cdot (\mathrm{ch}_Y \circ f_{!}) = f_* \circ (\mathrm{ch}_X \cdot \mathrm{td}_X)$. I am not sure how to phrase this as a naturality condition in the sense of category theory.

Solution 2:

Although I tend to subscribe to the Eilenberg-Maclane "thesis", I know of one example that behaves oddly: looking at Hilbert spaces over $\mathbb R$ (to avoid needing to think about complex conjugation), the Riesz-Fischer isomorphism $j_V:V\to V^*$ of a (real) Hilbert space and its dual $V^*$ (=continuous real-linear functionals on it) is not "natural...", in the sense that, for (continuous linear) $f:V\to W$, it is rarely the case that $f^*\circ j_W\circ f=j_V$. That is, the obvious squares do not commute.

(For example, if $f:V\to W$ is not injective, then the adjoint $f^*:W^*\to V^*$ is not surjective... Even more simply: for $f:\mathbb R\to \mathbb R$ by multiplication by $t\not=\pm 1$, the adjoint is multiplication by $t$, and $f^*\circ j_V\circ f=$ multiplication by $t^2$...)

In fact, I've never needed this (non-existent) "naturality", so it's not impeded anything I've been doing. Nevertheless, until just a few years ago, if someone had asked me whether the Riesz-Fischer map was "natural" (apart from issues about complex conjugation...), I'd have immediately said "yes".