I’ve been reading the Xena Project blog, which has been loads of fun. In the linked post Kevin gives the natural isomorphism $V to V^{ast ast}$ from a f.d. vector space to its dual as an example of a "canonical isomorphism":

People say that the obvious map from a vector space to its double dual is “canonical” but they could instead say that it is a natural transformation.

I think this is not a complete description. When we say that $V$ is canonically isomorphic to its double dual we don’t just mean that there *exists some* natural isomorphism from the identity functor to the double dual functor – what we mean, *and what we use in practice*, is that a *particularly canonical* natural transformation $V to V^{ast ast}$ is an isomorphism. We can multiply any such natural isomorphism by a scalar $c neq 0, 1$ in the ground field $k$ and obtain a different natural isomorphism between these two functors, and these are *not* canonical!

But reading the linked post showed me I don’t know precisely what I mean by "canonical" in the paragraph above. So:

Question:What, precisely, do we mean by "canonical" when we say that the usual natural transformation $V to V^{ast ast}$ is the "canonical" one we want? What other "canonical" maps are canonical in the same way?

What follows are some off-the-cuff thoughts about this.

One thought is that when we write this map down we’re "using as little as possible"; we’re not even really using that we’re working in vector spaces. If we define the dual to be the internal hom $[V, k]$ then ultimately all we’re using is the evaluation map $V otimes [V, k] to k$ together with currying. In other words, one way to make precise what we’re using is the closed monoidal structure on $text{Vect}$. (We don’t even need a closed structure if we define the dual to be the monoidal dual but I expect the internal hom to be more familiar to more mathematicians.)

So, here’s what I think ought to be true: in the *free* closed monoidal category on an object $V$ (blithely assuming that such a thing exists – maybe I should have used monoidal duals after all, because I’m much more confident that the free monoidal category on a dualizable object exists), there ought to be a *literally unique* morphism $V to [[V, 1], 1]$ where $1$ denotes the unit object. This is the "walking double dual map" and, by the universal property, reproduces all the other ones, and I think *this* is a candidate for what we might mean by "canonical map."

This trick of taking free categories is a really good trick actually, since it also cleanly describes the *nonexistence* of canonical maps. For example there is no canonical map $V to V otimes V$, and you could use representation theory to show that the only $GL(V)$-equivariant map is zero, but actually it’s just true that in the free monoidal category on $V$ there are no maps $V to V otimes V$ whatsoever. Similarly there is no canonical map $V to V^{ast}$, and similarly you could use representation theory to show that the only $GL(V)$-equivariant map is zero, but actually it’s just true that in the free closed monoidal category on $V$ (if it exists; take the free monoidal category with duals if not) there are no such maps whatsoever.

But I don’t think this sort of reasoning is enough to capture other examples where two functors are naturally isomorphic and there’s a particularly canonical natural isomorphism that we want in practice. For example, it’s also true that de Rham cohomology $H^{bullet}_{dR}(X, mathbb{R})$ on smooth manifolds is canonically isomorphic to singular cohomology $H^{bullet}(X, mathbb{R})$ with real coefficients, and by this we don’t mean just that there exists some natural isomorphism but that the *particularly canonical* natural transformation given by integrating differential forms over simplices is an isomorphism. We can multiply any such natural isomorphism by $c^n$ in degree $n$ where $c in mathbb{R} setminus { 0, 1 }$ and we’ll get a different one, even one that respects cup products, and again these are *not* canonical (although I wonder if we could easily distinguish the usual one from the one obtained by setting $c = -1$ – do we have to choose how simplices are oriented or something like that somewhere?) But now I really don’t know what I mean by that! It doesn’t seem like I can pull the same trick of zooming out to a more general categorical picture. de Rham cohomology is a pretty specific functor defined in a pretty specific way. Maybe this one is a genuinely different sense of "canonical," closer to "preferred," I don’t know.

Some previous discussion of "canonical" stuff on MO:

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