The meaning and purpose of "canonical''

Question

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:

1. 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.

2. 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.

3. 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.

4. 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.

5. 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?

Answer 1

The term "canonical basis" is used in representation theory. One of the fundamental examples is the Kazhdan–Lusztig basis of the Hecke algebra of a Coxeter group. Why is the term "canonical" used? Well, vaguely speaking, the thought process begins by trying to categorify the Hecke algebra. There's no mechanical recipe for categorification, and there are various categories that may be regarded as categorifying the Hecke algebra, such as (in the crystallographic case) the Bernstein–Gelfand–Gelfand category of representations of the associated complex simple Lie algebra, or a certain category of perverse sheaves on the corresponding flag variety. But once you have a category, you can get a distinguished basis by considering the classes of simple objects in the Grothendieck group. These give rise to the KL basis in a natural way.

Regarding your question of whether this use of "canonical" is absolute or relative, I would say that it is relative. On the other hand, the KL basis exudes an uncanny odor of being the "right" basis in some sense. Consider for example the positivity of the coefficients of the KL polynomials. The search for an explanation of this positivity has led to the discovery of all kinds of unexpected algebraic and (sometimes) geometric structure. For more information, see for example The Hodge theory of Soergel bimodules and The $p$-canonical basis for Hecke algebras.

Answer 2

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.

Answer 3

For Bourbaki, the adjective "canonical" is simply a way to assign a name to a well-defined object. For instance, when the quotient map $E\rightarrow E/R$ is defined ($E$ is a set, $R$ an equivalence relation), Bourbaki says "it is called the canonical map from $E$ to $E/R$"; then each time a quotient map occurs, Bourbaki will say "let us consider the canonical map $E\rightarrow E/R$ ... ", and it should be clear for the reader what it is.

Answer 4

I believe that there is no definition of canonical which covers all of its uses in mathematics. Heuristic evidence for this is that the definition of a canonical map in Wikipedia is vague.

The word appears in number theory and in some cases it can be made rigorous, and in other cases it is more of a heuristic issue.

Sometimes people say "canonical isomorphism" when they just mean "natural (in the sense of category theory) isomorphism".

In algebraic geometry there is a convention where canonically isomorphic objects are declared equal and the $=$ symbol is used to denote the canonical isomorphism. Strictly speaking this is abuse of the $=$ sign but it has never seemed to cause any problems in algebraic geometry. To give an explicit example: if $R$ is a commutative ring and $M$ is an $R$-module then there is a canonical identification of the abelian groups $M_f:=M\otimes_R R[1/f]$ and $M_g:=M\otimes_R R[1/g]$ when the set of primes containing $f$ equals the set of primes containing $g$ -- both of these are the value on the open subset $D(f)=D(g)\subseteq Spec(R)$ of the quasicoherent sheaf associated to the $R$-module $M$. In EGA1, (1.3.3) Grothendieck says that the map between these modules is "un homomorphisme canonique fonctoriel" and then uses the $=$ symbol and writes $M_f=M_g$. They are not equal in the strict sense, but they both satisfy the same universal property so as long as mathematicians only ever do things to these modules which can be done using the universal property -- and this is exactly what they do -- then this abuse of the $=$ symbol will not cause problems. But one could in theory argue that many many uses of the $=$ symbol in EGA and elsewhere in algebraic geometry should really say "is canonically isomorphic to". In Milne's book on etale cohomology he explicitly admits this, saying at the beginning of his book that canonical isomorphisms will be denoted $=$, without ever explicitly saying what a canonical isomorphism is. Here, what appears to be going on is that the isomorphisms are sufficiently natural that every conceivable sensible diagram that one can come up with will commute. Note that there is an extra subtlety here -- $A = B$ is usually in mathematics a true-false statement. In Milne's book it is sometimes actually implicitly encoding a map from $A$ to $B$, and the usual proof that equality is an equivalence relation is being beefed up to assertions of the form "composing two canonical isomorphisms gives a canonical isomorphism" and so on. This phenomenon occurs all over the place and is very well-hidden. However there is no doubt that this convention "works".

Finally, in the Langlands philosophy one can find assertions saying that the local or global Langlands correspondences are canonical. This is a use of the term which I find more unsettling because the situation seems to be that people believe that there is one "special" correspondence, and they can list a bunch of properties which it should have, but nobody has proved the theorem that there is at most one correspondence with these properties and indeed it may well not be true that the current list of properties that we have defines the correspondence uniquely. One can try and wriggle out of this by asking that the correspondence commutes with functoriality, however functoriality is a big open question and the statement of functoriality is not completely understood in full generality (what happens for non-classical groups at bad primes). My current thinking about this use of the word is that it is more expressing a hope that one day in the future, people will figure out what we were talking about. However if one works with general connected reductive groups, I am pretty sure that this day is not yet here. It is for me a very interesting usage of the word.

Finally it might be worth mentioning that even in the case of GL_1, there are two canonical Langlands correspondences! Class field theory is the statement that two abelian groups are canonically isomorphic, but these groups contain elements of order bigger than 2, and so one can apply inversion on one side but not the other and change an isomorphism into a different one. They are distinguished by having different names -- one is "the canonical isomorphism sending uniformisers to arithmetic Frobenii" and the other is "the canonical isomorphism sending uniformisers to geometric Frobenii".

Answer 5

This seems broader than the other question which was interpreted mainly in the “category theory” sense (1.).

An early, maybe earliest,^a case of sense (4.) “normal form” is Jacobi (1837) calling canonical the “Hamilton form”^b of the equations of mechanics, and also any variables, coordinates or “elements” in which they take this form; today we would speak of Darboux coordinates or Darboux normal form^c of a symplectic structure. Remarks:

1. A big difference with the case of fractions is that the normal coordinates (or isomorphism to normal form) are far from unique.

2. Sylvester in another context (1851, p. 190) attributes the phrase to Hermite, presumably (1854):

   I now proceed to the consideration of the more peculiar branch of my inquiry, which is as to the mode of reducing Algebraical Functions to their simplest and most symmetrical, or as my admirable friend M. Hermite well proposes to call them, their Canonical forms.

3. Similar: the Jordan canonical form^d and Frobenius rational canonical form^e of a matrix.

4. So far as I can tell, Jacobi may well have originated the phrase “normal form” too (1845, 1850).

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^a one can find “ad formam canonicam” at least once in Euler: De reductione formularum integralium ad rectificationem ellipsis ac hyperbolae (1766, p. 28); also often aequatio canonica.
^b notoriously used before by Lagrange and Poisson.
^c called canonical by Frobenius (1877), canonical or normal by Darboux (1882).
^d called canonical by Jordan (1870, p. 114); Kronecker (1874) bitterly deplored the terminology.
^e called normal by Frobenius (1879, pp. 207-208).

Answer 6

Every group (actually I believe the same holds for rings and modules) admits the following (silly) canonical presentation. If $G$ is a group, consider the free group $F(G)$ on the (underlying) set $G$. Then $$G = F(G)/R,$$ where $R$ is the subgroup normally generated by words of the form $ghk^{-1}$, where $g$, $h$ and $k$ are in $G$ satisfying $gh=k$.

Answer 7

(a) Does the term 'canonical' appear in your field? If so what is the sense of the term? Is it relative or absolute?

Practical algorithms for the graph isomorphism problem solve the graph canonization problem instead of merely checking two given graphs for isomorphism. Normally this is done by computing a canonical labeling, but other canonical forms would serve the same purpose, namely to allow efficient set/map operations (in the sense of C++ std::set/std::map) after the given graphs have been canonized.

The sense of the term here seems to be to single out a unique representation or hash for a given graph. I would say the term is relative here, i.e. it is not important to determine the "best canonization," and it isn't even clear what "best" would mean in this context.

(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?

The graph canonization problem is slightly more difficult than the graph isomorphism problem. So after László Babai showed that graph isomorphism can be solved in quasipolynomial time, it was clear that one should try to also find a corresponding canonization algorithm. So László Babai wrote a follow up report Canonical form for graphs in quasipolynomial time: preliminary report and others tried to address the problem from a more general point of view (like Pascal Schweitzer and Daniel Wiebking in A unifying method for the design of algorithms canonizing combinatorial objects)

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Let me also mention a different example where I used the term canonical myself. The straightening theorem for vector fields says that around a point where the vector field is non-zero, there exist a local coordinate system in which the vector field is constant. But this coordinate system is not unique. To get a unique straightening, I extended the vector field by an additional coordinate, and set the vector field there to 1. This allows me to use a unique straightening, which I call the canonical straightening. I used this canonical straightening for short proofs of some theorems, and I believe that it made the arguments more transparent.

I also extended that trick to the case of multiple vector fields whose Poisson brackets (or maybe it is called Lie bracket in this case) form a nilpotent Lie-algebra. Here I extend with more additional coordinates, set the vector field there to a simple presentation of the nilpotent Lie-algebra, and get a coordinate system (in a "canonical" way) where the vector fields are given by that simple presentation. That was done in a German text. I never came around to investigate whether that trick could also be made to work for solvable Lie-algebras, or whether something similar could even be done for arbitrary Lie-algebras.

My use of canonical in the last example above (where I use some simple presentation of a nilpotent Lie-algebra) doesn't feel significantly different to me than the use of canonical in the graph canonization problem. So it is neither a "best" nor "universal" construction, but just a sufficiently unique construction.

Answer 8

Most people who study mathematics beyond high school level would be introduced to the word canonical in Algebra I. At first, a set of vectors $\{e_1,e_2,\ldots ,e_n\}$ is defined as basis if it is linearly independent and spans the whole space. How ever, that set is not unique to a specific space, so we introduce canonical basis with Kronecker delta $$e_i=(\delta_{1i},\delta_{2i}, \ldots ,\delta_{ni}).$$ What does the word canonical mean in this context? I would say a way to pick out a unique basis, that is standard and can be applied in practice. More interestingly, is it chosen arbitrarily or is it inherent for us, humans, to think of space in terms of up, left, right?

Answer 9

Given Francois Ziegler's answer, my interpretation here may be ahistorical, but here's how I think about the use of "canonical" in the context of symplectic geometry/mechanics.

If you have a phase space (what we'd now call a symplectic manifold) and you pick half of your coordinates (say those corresponding to position $q_1,\ldots,q_n$) then the other half come for free (the canonically conjugate momenta): the functions $q_1,\ldots,q_n$ generate Hamiltonian vector fields $X_{q_1},\ldots,X_{q_n}$ and the "$p_k$ momentum axis" is simply the flowline of $X_{q_k}$. Indeed, this is one way you construct Darboux coordinates (e.g. in Arnold's book on classical mechanics).

Answer 10

Some of the examples given in the question are a careless misuse of the word. Who writes "canonical basis" for $K^n$ when they mean "standard basis", and who writes of a "canonical presentation of a fraction" when they mean a "fraction in its lowest terms", which isn't even canonical unless everything is positive.

In my field, arithmetic geometry, "canonical" has a well-understood meaning even if it is difficult to write down a precise definition. In his 1980 book, Milne was comfortable assuming that his readers would know what it meant (in his later writings, he has switched to using $\simeq$ for "canonically isomorphic"). Roughly, it means that the object can be constructed without making any arbitrary choices. There is a huge difference between saying two objects are isomorphic and saying they are canonically isomorphic. Barr botched this in his translation of Grothendieck's Tohoku paper by replacing "=" (meaning canonically isomorphic) with isomorphic.

I agree that the use of "canonical" is problematic in the Langlands program. There are major conjectures saying that some set (of representations) is bijective to some other set. After Serre pointed out that this only means that the two sets have the same cardinality, the word "canonical" was added. It is part of the problem to figure out what that means.