This question is jointly formulated with Neil Barton. We want to know about the significance of canonicity in mathematics broadly. That is, both what it means in some detail, and why it is important.
In several mathematical fields, the term 'canonical' pops up with
respect to objects, maps, structures, and presentations. It's not
clear if there's something univocal meant by this term across
mathematics, or whether people just mean different things in different
contexts by the term. Some examples:
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In category theory, if we have a universal property, the relevant
unique map is canonical. It seems here that the point is that the map
is uniquely determined by some data within the category. Furthermore, this kind of scheme can be used to pick out objects with certain properties that are canonical in the sense that they are unique up to isomorphism. -
In set theory, L is a canonical model. Here, it is unique and definable. Furthermore, its construction depends only on the ordinals– any two models of ZF with the same ordinals construct the same version of L.
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In set theory, other models are termed 'canonical' but it's not
clear how this can be so, given that they are non-unique in certain
ways. For example, there is no analogue of the above fact for L with respect to models of ZFC with unboundedly many measurable cardinals. No matter how we extend the theory ZFC + "There is a proper class of measurables," there will not be a unique model of this theory up to the specification of the ordinals plus a set-sized parameter. See here. -
Presentations of objects can be canonical: The most simple being
that of fractions, whose presentation is canonical just in case the
numerator and denominator have no common factors (e.g. the canonical
presentation of 4/8 is 1/2). But this applies to other areas too; see here. -
Sometimes canonicity seems to be relative. Given a finite-dimensional vector space, there is a canonical way of defining an isomorphism between V and its dual V* from a choice of a basis for V. This determines a basis for V*, and thus the initial choice of basis for V yields a canonical isomorphism from V to V**. But two steps can be more canonical than one: The resulting isomorphism between V and V** does not vary with the choice of basis, and indeed can be defined without reference to any basis. See here.
Other examples can be found here.
Our soft questions:
(a) Does the term 'canonical' appear in your field? If so what is the
sense of the term? Is it relative or absolute?
(b) What role does canonicity play in your field? For instance, does it help to solve problems, help set research goals, or simply make results more interesting?
Best Answer
In my experience "canonical" means "the simplest way possible" within some context. Often it turns out that this way is uniquely determined, at least when one puts some "natural" restrictions.
For instance, when we want to embed a domain $R$ into its field of fractions $Q(R)$, the simplest way to do that is to map $r$ to $\frac{r}{1}$. All other formulas either don't work, or they don't define a ring homomorphism, or they are not injective. And this is actually the only embedding which can be defined for every domain $R$ and is a natural transformation.
Another example is the projection map $X \times Y \to X$, $(x,y) \mapsto x$. Again this is the simplest way to produce an element of $X$ out of an element of $X \times Y$. And this is actually the only choice which is natural in $X$ and $Y$.
The canonical basis of $K^n$ is another example. Here the simplicity is measured by the number of zeroes in each basis vector, and zeroes should be considered to be simple of course.
In order to illustrate that uniqueness is not required in general, for example one says that for sets $X$ there are two canonical maps $X \to X \sqcup X$. Likewise, there are two canonical maps $X \times X \to X$.
In some cases there is no canonical solution. For example, I would argue that there is no canonical bijection $\mathbb{N}^2 \to \mathbb{N}$, and in fact I don't see a clear measure for simplicity here. Cantor's pairing function is a polynomial bijection which can therefore be considered to be quite simple, but this is just one choice among many others. And one could argue that $(n,m) \mapsto 2^n \cdot (2m+1)-1$, even though it's not polynomial, is actually much simpler since here bijectivity is trivial.
One purpose of canonical maps, structures etc. is to focus on what is relevant and useful.