I find non-canonical isomorphisms very interesting, but I wish I knew more examples. To be vague, an isomorphism (perhaps in a category) is said to be non-canonical if it requires making an “arbitrary choice.” One of the reasons I find them interesting is that we often think of objects only up to isomorphism, but in order for certain things to make more sense we must distinguish between objects that are non-canonically isomorphic. Here are the examples I know of.
Vector space duality
Finite-dimensional vector spaces are non-canonically isomorphic to their duals. This is one of the simplest examples to explain, but I think it is often ignored in elementary linear algebra courses. The dual of a vector space is the space of linear functionals to the base field, and of course by linearity a linear functional is uniquely determined by what it does to each element of a basis; in other words, numbers. So all we have to do is pick a basis and we can write down the space of linear functionals, which looks just like the original space.
But this is just an illusion. Here’s the simplest way I can think of to clarify the difference: if vector spaces are column vectors, then their duals are row vectors.
Usually no distinction is made between row and column vectors because when we write vectors down numerically (which, you’ll note, also requires a choice of basis), taking transposes takes us from one to the other and they both look pretty much the same. But matrices (that is, linear transformations) don’t act the same on row and column vectors! A given square matrix acts on column vectors from the left and row vectors from the right, and these actions are different unless the matrix is symmetric. Of course, there’s also a transpose operation here, but what becomes very clear now is that the transpose operation is basis-dependent, so until we learn to talk about it in an abstract way we shouldn’t trust what it does numerically.
So here’s the abstract language: if we call column vectors “vectors,” then row vectors aren’t really vectors – they’re called covectors, which is another word for dual vectors. Covectors don’t behave the same as vectors because they transform covariantly under change of basis, whereas vectors transform contravariantly. This is a fancy way to distinguish the left and right action of matrices. This distinction is important in mathematical physics, but I don’t have the background to really discuss why in detail.
Here’s why I like covectors: they do a good job of motivating the dot product and the tensor product. There is a natural pairing of vectors and covectors (it’s just evaluation) which is bilinear, and a dot product arises once you identify the dual space with the original space (non-canonically, of course). Generalizing instead of specifying, there is a universal way to talk about bilinear maps called the tensor product (another thing often ignored in linear algebra courses), and combining it with duality allows us to think of certain constructions in various equivalent ways. For example, a linear operator is uniquely specified by a list of vectors that we assign to each element of a basis. If we change basis, this list of vectors changes covariantly – so it’s really a list of covectors! More formally, the following is true.
Proposition: The space of linear maps is canonically isomorphic to the tensor product .
This isomorphism is called canonical because, while it can be defined by an arbitrary choice of basis, the final isomorphism does not depend on the choice of basis. So now we can explain the transpose operation: to find a transpose, write a linear operator as an element of after picking a basis, and then switch vectors and covectors.
I didn’t finally get around to learning all of this stuff until I read Linear Algebra via Exterior Products by Sergei Winitzki, which is freely available at the URL; you can find all the details of what I’ve been talking about there. I found it an extremely interesting supplement to what I already knew about linear algebra; in particular, it “explains” the determinant formula in a way that I haven’t seen a standard linear algebra text try to tackle (via, as the title suggests, the exterior product). Rota once wrote that Noether’s contributions to the study and development of abstract algebra is the reason we learn Galois theory instead of tensor algebra. I wonder if this is necessarily a good thing.
Permutations and total orders
On finite sets, permutations are non-canonically isomorphic to total orders. Now, I have to be somewhat careful what I mean when I say this: one way to make this precise is that they have the same generating function (there are permutations and total orders on a set with elements), but as combinatorial species they are not related by a natural transformation.
So what’s the non-canonical isomorphism (that is, bijection)? Actually, there are quite a lot of them (which is sort of what “non-canonical” means in the first place!). For example, one representation of a permutation is its “word,” which is the sequence describing the image of each element of the underlying set. Reading the word from left to right gives you a total order on the set. Easy enough, right? But until we name the elements of the set, they are anonymous: labeling them is the non-canonical choice being made. (In fact, it’s a non-canonical choice of total order!)
Another representation of a permutation is its cycle decomposition. There is a unique way to write down a cycle decomposition so that each cycle begins with the smallest element and the cycles themselves are ordered by their smallest elements (which, again, requires a non-canonical choice of total order), and reading a cycle decomposition as a word gives us another permutation, or another total order. Stanley calls this the “fundamental bijection.”
Stanley also discusses a fascinating q-analog of the connection between permutations and total orders relating complete flags in vector spaces over finite fields (the analog of total orders) to the corresponding linear groups (the analog of permutations) that I haven’t yet digested.
Vector spaces are non-canonically isomorphic to their underlying affine spaces. Like the relationship to dual spaces, the distinction between affine and vector spaces is occasionally ignored, which only causes confusion. There is a very nice generalization of this relationship that goes by the name of a torsor associated to a group. A torsor for a group is a G-set where the action is both transitive and free, meaning for any two elements in the underlying set there is exactly one group element such that . Thus, while it is not possible to “add” elements of the set, one can “subtract” two set elements and get a group element, or “add” a group element and an element of the set to get a set. As Baez explains so well in the link above, this relationship shows up everywhere in physics, and the distinction between affine and vector spaces is just a special case. The non-canonical choice here is the choice of an identity or origin (in the affine case): if we fix some element , then we can identify any other element of the set with the unique such that . But often we don’t want to make this non-canonical choice; as Baez notes, a simple example is the usual way we write down antiderivatives.
Sometimes vectors are defined in terms of affine spaces. Take the difference between two points and get a vector; add a point and a vector and get a point. Confusingly, people often write down both vectors and points using coordinates, but there are two non-canonical choices involved in writing down points as coordinates: you must choose both a basis and an origin.
Torsors are very cool. They even generalize the second example: the set of total orders on a set with elements is nothing other than an -torsor. The choice of an “identity” total order (as we saw above) provides the non-canonical isomorphism. But if we use the torsor language, we don’t have to make any such choices; we don’t even have to name the elements of the set!
Is there a concept that generalizes both “torsority” and “duality”? What other non-canonical choices are common in mathematics?
For example, there is the notion of Hodge duality, which depends on a non-canonical choice of orientation and orthonormal basis, but in some sense Hodge duality is just a fancier vector space duality. Explicitly, given an orthonormal basis of a vector space the wedge product defines a bilinear pairing , and since is one-dimensional (it is spanned by ), identifying it with (the simplest case of Hodge duality) gives the other cases of Hodge duality immediately. Hodge duality is fun because it explains two other mysteries of basic linear algebra: determinants and cross products (and the relationship between the two).
Perhaps a sufficiently general version of duality does the trick. Given a pairing , one can either think of elements of as parameterizing functions or think of elements of as parameterizing functions ; these perspectives are “dual.” We get vector space duality by setting , with the pairing given by evaluation and “torsority” by setting with the pairing given by the group action. The non-canonicity arises by picking certain elements of . But I can’t help but feel like there’s more to the picture, and I can’t currently think of any genuinely distinct examples.