What is the definition of "canonical"?

Question

I just received a referee report criticizing that I would too often use the word "canonical". I have a certain understanding of what "canonical" should stand for, but the report shows me that other people might think differently. So I am asking:

1. Is there a definition of "canonical"?

2. What are examples where the use of "canonical" is undoubtedly correct?

3. What are examples where the use of "canonical" is undoubtedly incorrect?

VERY LATE EDIT: I just came across this wonderful passage written by R.Lipschitz, In a correspondence with André Weil (Oeuvres, vol. 2, page 558):

I can assure you, at any rate, that [...] my results are invariant, probably canonical, perhaps even functorial.

Answer 1

I always had the following working definition of canonical (which I think Gordon James told me and he might have said it was due to Conway? Not sure): a map $A\to B$ is canonical if you construct a candidate, and the guy in the office next to you constructs a candidate, and you end up with the same map twice.

Somehow there is something more to it than that though. For example if $A$ is an abelian group and we want a map $A\to A$ then I will choose the identity, but I know for sure that the wag in the office next door to me will choose the map sending $a$ to $-a$ because that's his sense of humour. What has happened here is that there are in fact two canonical maps $A\to A$. This issue shows up in class field theory, which is an isomorphism between two rather fancy abelian groups $X$ and $Y$, and where no-one could decide for a long time which one of the two canonical isomorphisms was "best". So you often see statements in number theory papers saying "we normalise our class field theory isomorphisms so that geometric Frobenii go to uniformizers" (the alternative being the inverse of this). It also shows up in the Weil pairing on an elliptic curve: it's canonical, but because we're in an abelian situation, its inverse is too. So you see in e.g. Katz-Mazur an explicit spelling out of which of the two canonical choices one is going to make (and hang all the non-canonical ones!).

Answer 2

I think there is a multi-level classification associated to "canonicalness," which explains why some clashes of definition occur.

• Arbitrary — No requirements.
• Uniform — There may be a few options but these options can be selected by making a few global choices.
• Canonical — As in the uniform case, but there is only one natural choice of options which applies globally.

Canonical examples à la Russell:

• Choose one sock from each pair in a collection of sock pairs — There is no way to make a uniform choice.
• Choose one shoe from each pair in a collection of shoe pairs — There are two obvious global solutions, left shoe or right shoe, but no way to prefer one over the other.
• Choose one object from each set in a collection of sets each consisting of a bowtie and possibly other items — There is only one obvious global solution.

I think the main point of contention is distinguishing uniform and canonical. Some will argue that it's not canonical if there is a choice to be made, while some will argue that a finite number of global choices is still canonical.

There is yet another use of canonical to mean something like 'universally sanctioned' (this is closer to the religious term). The second occurrence of canonical above is of this type.

Answer 3

1. Not a definition, exactly; I would say the situation is similar to that of forgetful functor. If I say there is a canonical isomorphism between X and Y, then what I mean is that if asked, pretty much everyone would choose the same isomorphism. A canonical isomorphism is very often a natural isomorphism in the sense of category theory, but the converse need not hold. A canonical isomorphism does not need to be the unique isomorphism between X and Y, though sometimes it is when X and Y are considered as equipped with some additional structure.

2. "There is a canonical isomorphism between the set of elements of a ring R and the set of ring maps $\mathbb{Z}[x] \to R$." Obviously, I mean for $r \in R$ to correspond to the ring map sending $x$ to $r$, although I could just as well send $x$ to $-r$.

3. "There is a canonical isomorphism between a finite-dimensional vector space V and its dual." No explanation needed, I suppose.

Maybe more interesting would be an example where the word "canonical" is arguably correct or incorrect; I can't think of one off-hand.

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Addendum, after reading some of the other answers: I would emphasize that for me there is a difference between "natural" in the formal category-theoretic sense and "canonical". For one thing there is a linguistic distinction: if I am considering an isomorphism F between X and Y then "Theorem: F is a natural isomorphism" is perfectly acceptable but "Theorem: F is a canonical isomorphism" is very strange to me. There should be only one canonical isomorphism between two things, though what that isomorphism is could depend on context, e.g., "the canonical isomorphism $A \otimes B \to B \otimes A$" where $A$ and $B$ are graded abelian groups might mean different things to an algebraic geometer and an algebraic topologist.

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Finally, this is hardly a definition, more of a rule of thumb: there is a canonical isomorphism between X and Y if and only if you would feel comfortable writing "X = Y".

Answer 4

For me the word “canonical” always means “functorial in some sense”, usually without using any form of the axiom of choice. For example, every finite-dimensional vector space is canonically isomorphic to its double dual, because there is an isomorphism of functors id → **, but there is no canonical isomorphism between a finite-dimensional vector space and its dual, because one cannot construct an isomorphism of functors id → * without using some form of the axiom of choice. Likewise, the construction of an algebraic closure is not canonical because there is no functor that sends a field to its algebraic closure, even though every two algebraic closures are (non-canonically) isomorphic.

I presume that one can allow using the axiom of choice and still get the same results, but in this case one needs to use the language of 2-categories. For every well-pointed elementary topos T (basically, a set theory), we can construct the category of finite-dimensional vector spaces in this topos and isomorphism of functors id → **. I think that this isomorphism depends 2-functorially on T. On the other hand, even if we use the axiom of choice to construct an isomorphism of functors id → * for every well-pointed elementary topos T, there is no way to make it depend functorially on T. I must say that I have never tried to prove any of these statements, so they might as well be totally wrong.

Answer 5

I was taught to think that there is a precise definition of "canonical" in differential topology, at least in the context of linear algebra constructions. A construction is canonical if it is a smooth functor. (There is a Wikipedia page about smooth functors but it is not very insightful). And since it is hard to invent a non-smooth functor, it practically boils down to just being a functor.

The categories involved are usually not mentioned explicitly, and they are not things like vectors spaces with linear maps. They are rather things like vector spaces with linear isomorphisms as morphisms. Or, more generally, isomorphisms of whatever structure you happen to have on them. For example, dual vector space is a canonical construction but an isomorphism between a vector space and its dual is not. On the other hand, there is a canonical one if your spaces carry Euclidean structure.

The idea is that a canonical construction can be applied fiber-wise in fiber bundles. Sometimes this feature is advertised as a poor man's definition of "canonical" but this is not quite correct: for example, every vector bundle (over a paracompact base) is isomorphic to its dual, but this is not really canonical.

Answer 6

On page vii of the Introduction to the 1996 edition of Sheaf Theory by Glen E. Bredon, the author discusses the difference between "canonical" and "natural" and points to a historical context:

Occasionally, we use the equal sign to mean a "canonical" isomorphism, perhaps not, strictly speaking, an equality. The word "canonical" is often used for the concept for which the word "natural" was used before category theory gave that word a precise meaning. That is, "canonical" certainly means natural when the latter has meaning, but it means more: that which might be termed "God-given." We shall make no attempt to define that concept precisely. (Thanks to Dennis Sullivan for a theological discussion in 1969.)

Answer 7

I have two competing interpretations of the word canonical.

One, apparently the one used by Bourbaki, is mathematically informal. In various contexts, some objects (maps, modules, etc.) are defined unambiguously and called canonical. For example, the canonical basis of the free module $A^{(I)}$ over a set $I$, the canonical surjection from a set $X$ to its set $X/R$ of equivalence classes with respect to some equivalence relation $R$, the canonical bilinear map from a product of two modules to their tensor product, etc.

The other interpretation is categorical. The given context defines (often implicitly) some categories and the canonical object is functorial with respect to isomorphisms. It is more or less what Kevin Buzzard says in its answer, when he defines canonical by the property that he and a colleague, when asked to define the object, would agree on the same object.

Answer 8

I was always under the impression that canonical meant, precisely, that no arbitrary choices were necessary. But, that it was occasionally used less formally, in a more standard-English sort of way to mean traditional/obvious/well known. The informal meaning is usually used as a cheap way to avoid explaining something that's easier for the reader to guess anyway.

Ex 1: Two vector spaces of the same dimension are isomorphic. The isomorphism is not canonical.

Ex 2: A finite dimensional vector space is canonically isomorphic to its double dual.

Ex 3: Let $\pi: S^3 \to S^2$ be the canonical fibration.

I never really liked it when people use canonical as in example 3. It seems like using it this flexibly detracts from the useful technical interpretation of the word.

I've also heard some more complicated category theoretic interpretations of what canonical meant. But, after more scrutiny, it seems that these "definitions" are specific cases of the "no arbitrary choices" principle.

Answer 9

I would say that "canonical" ought to be used to describe when no choices have been made.

A nice example of a non-canonical identification: A principal bundle is made up of principal homogeneous spaces for the action of a Lie group. These are spaces which are homeomorphic but non-canonically isomorphic to the Lie group. For example, I might have a circle bundle. My Lie group would be a `concrete version' of the group such as $\{|z| = 1\}$, but my fibres are simply circles. I would need to choose a base point on each of the circles to make them into groups in the same way. This amounts to taking a global section and can't always be done (e.g. circle bundle on the sphere has no global section by hairy ball theorem), so the non-canonical-ness might actually be important

The labelling of identifications as Canonical and Non-canonical is common in linear algebra: Since one chooses bases so often, it is worth pointing out when such a choice is avoided...

To prove that $V^* \otimes V^*$ is isomorphic to $(V \otimes V)^*$, one ought to work with elements of the spaces directly rather than their representations in some basis. I would therefore call the resulting isomorphism `canonical'.

Answer 10

Not a definition, but an example of use in logic:

In model theory, "canonical" is often used in the phrase "the canonical model" to mean "intended structure." For instance, in first-order logic, one may speak of "the canonical model of Peano Arithmetic" to mean the structure of the natural numbers, or "the canonical model of the theory of real-closed fields" to mean the field of real numbers. Intuitively, "the canonical model" of a theory is the structure one was trying to pin down when the axiomatisation of the theory was written. It's just that in first-order logic, it is hard to pin down (infinite) structures! No first-order theories admitting infinite models are categorical (they admit non-isomorphic models; indeed, they admit models of every infinite cardinality), and compactness/ultraproduct/(many other) constructions can often be used to build "non-standard" models of theories. "Non-standard" models of Peano Arithmetic or the theory of real-closed fields would in this context be called "non-canonical" (even though there are many canonically studied "non-standard" models of those theories!).

But, many commonly studied theories do not have a notion of "canonical model." For instance, one would not say "the canonical model of group theory."

Answer 11

Vague definition of canonical:

Let $X$ and $Y$ be collections (often sets) for which assumptions has been made (has been given structures and/or are related somehow). A function $f\colon X \to Y$ is canonical if it is given by a rule using only the already given structure.

This explains the relation to the greek word rule (kanon). The precise meaning of the above are open for enterpretation: how much structure can the rule itself contain (maybe this can be made precise)! This "definition" somewhat contradicts many of the other answers, which for some reason is under the impression that canonical implies unique (or almost unique), which in my point of view is very wrong since different rules may define different maps. E.g. if we let $X$ be the objects in the category of abelian groups and $Y$ the morphisms then the definitions makes all the group homomorphisms $A \to A$ given by multiplication with an element in $\mathbb{Z}$ canonical, which to me is not a problem.

Usually when there is an especially simple rule it is often assumed without mentioning that this is the rule defining the function. E.g. most will understand the following:"there is a canonical endemorphism of any object in a category". This emphasizes the multiplication with 1 above as somehow speciel or "more canonical" than the rest. This is simply because the rule works in much greater generality and is shorter.

Usually if a rule is very simple the function will have nice properties. E.g. simply rules in category theory often define functors, natural transformations, e.t.c. This leeds many people to confuse the notion of canonical with "something behaving nicely".

I am somewhat puzzled by the use of the word uniform in one of the answers. The nature of the word uniform is "of the same form" and relates more to symmetries and things looking the same every where. This often leeds to canonical maps, since a choice at one point can sometimes be extended to a choice at every point. Please someone comment on this since maybe this is just a use of the word I have not seen before!

Answer 12

Regarding the (widely considered to be false) statement that "There is a canonical isomorphism between a finite-dimensional vector space V and its dual" in Reid Barton's answer, I think that the situation is a bit more interesting than that. It is a good illustration of the idea that an object may be defined to be "canonical" if it is constructed without making any choices, and the interesting point here is that there are various degrees of (in-)tolerance to choices. If we work with vector spaces of fixed finite dimension, then an isomorphism $i_E:E\to E^*$ between a vector space $E$ and its dual $E^*$ may be called canonical if

1. it does not depend on the choice of a basis for $E$ but we need a basis to define it, or
2. it does not depend on the choice of a basis and may be defined without choosing a basis, or
3. it does not depend on basis choices as above and does not even depend on $E$, in the sense that whenever $u:E\to F$ is an isomorphism between dim. vector spaces of the same finite dimension, then $u^*\circ i_F\circ u=i_E$ where $u^*$ is the transpose of $u$. This third notion of canonicity is essentially functoriality.

Concretely, given a basis $B=\{e_j\}$ of $E$ with dual basis $B^*=\{e_j^*\}$, we can construct an isomorphism $i=i_{E,B}:E\to E^*$ that maps $e_j$ to $e_j^*$. This map does not depend on the choice of basis if and only if $u^*\circ i\circ u=i$ for all $u\in \text{GL}(E)$. It is easily seen that this is equivalent to the fact that $\text{GL}_n(k)=\text{O}_n(k)$, where $k$ is the base field, $n$ is the dimension, and $\text{O}_n(k)$ is the orthogonal group of the standard (sum of squares) quadratic form. Exercise: this equality holds if and only if $n=1$ and $k$ has at most $3$ elements. Thus for $n=1$ and $\text{card}(k)\leq 3$ the map $i_{E,B}:E\to E^*$ does not depend on $B$, so we may write it simply $i_E$. When $\text{card}(k)=2$ this is not so surprising because any two one-dimensional vector spaces over the field with two elements are uniquely (hence canonically in whichever sense you like) isomorphic, but for $\text{card}(k)=3$ this is a bit more exotic. Having reached this point, we might think that we are in the funny notion 1 of canonicity (and this is what I thought some minutes ago). But in fact, still assuming that $n=1$ and $\text{card}(k)\leq 3$, we can exhibit an isomorphism $i:E\to E^*$ without any reference to a basis. Namely, define $i(0)=0$ and if $x\in E$ is nonzero, then it is a basis of $E$, and we can define $i(x)=x^*$, the only element of the dual basis. The point is that since $a^2=1$ for all nonzero scalars in $k$, this map $i$ is linear.

Conclusion: if $n=1$ and $\text{card}(k)\leq 3$ there is an isomorphism $i:E\to E^*$ that is constructed without a choice of basis, and it is functorial for isomorphisms of one-dimensional vector spaces. If $n\ge 2$ or $\text{card}(k)\ge 4$, the map $i_{E,B}:E\to E^*$ is not independent of the basis $B$.

I would guess that it is possible to find examples of phenomena like 1 above.

Answer 13

Since this question has just popped up again to the the top of the stack, I can't resist adding one more response.

It occurs to me that constructive logic (or type theory) suggests a way to formalize the meaning of "canonical". The idea is that when you say "the canonical $x$ such that $P(x)$" (or "the canonical $x$ of type $P$"), your statement is implicitly justified by some proof that $\exists x . P(x)$ (or $\exists x : P$). If this proof is constructive, then it actually constructs a specific $x$ such that $P(x)$ or term $x$ of type $P$. When you say "the canonical $x$ ...", you "extract" that constructed $x$.

Whenever we talk about formalizing math, there must be some notion of "implicit" parts of a mathematical argument which must be reconstructed in order to formalize it. You might think that the implicit parts consist simply of implicit proofs, but the use of the word "canonical" actually allows us to leave implicit certain constructions. What the constructive logic (or type theory) does is to view all the implicit stuff, proofs and constructions alike, in the same light, since it treats a construction as a kind of proof.

It's also interesting that normally, if an argument relies on an implicit proof, then it doesn't matter what proof the reader supplies, so long as it's correct. But if your argument relies on an implicit construction witnessing an existence statement, and if you want to refer back to this construction as "the canonical $x$...", then it does matter what proof is supplied insofar as the proof has to construct the right witness. Maybe this starts to point toward homotopy aspects of the type theory involved?

Answer 14

I remember this exchange from a lecture by Prakash Panangaden:

"And so every quantum algorithm can be written in this canonical form."

"What do you mean by canonical?"

"Canonical means I like it a lot."

Answer 15

Hopefully I don't say something too stupid. I just wonder whether the definition of canonical might be relative.

For example, if we look at $ \mathbb Z / p$ as an additive group in fact there is no non-zero element which stands out. But if we look at $ \mathbb Z / p$ as a field $1$ stands out as a non-zero element.

Another example. From the geometrical reason alone, there is no good reason to choose a positive direction (Essentially there is no way to distinguish from left hand and right hand). But in a universe where there is electro magnetic force, we then have a canonical way to choose a positive direction.

Yet another example, there is a canonical way to choose whether you want a left shoe or a right shoe: If you are left-handed then choose the left one, if you are right handed choose the right one.

Perhaps what counts as canonical depends on where we are standing. A suggestion for a heuristic definition: canonical is definable with respect to the structure you are standing at.

Answer 16

For me, if we have a partition P of a set S, then we can define a set of representatives, one from each part of P, each of which is called canonical.

Typically, the partition P is formed by the orbits of a group G acting on S. If we choose G so that every element in S has a trivial stabiliser, then we can find |S| by instead counting the canonical representatives since |S|=|G|*|P| by the Orbit-Stabiliser Theorem.

Often, the elements that are chosen to be canonical can be quite contrived - e.g. just because your program outputs a certain element of P first, i.e. "lexicographical order".

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To add some examples:

a) An orthomorphism of $\mathbb{Z}_n$ is a permutation $\sigma$ such that $i \mapsto \sigma(i)-i \pmod n$ is also a permutation. We partition the orthomorphisms of $\mathbb{Z}_n$ into equivalence classes under the transformation $E_g$ for which $E_g\[\sigma\](i)=\sigma(i)+g \pmod n$. Therefore the parts each have cardinality $n$ and we define the canonical representatives to be the orthomorphisms $\sigma$ for which $\sigma(0)=0$. Therefore the total number of orthomorphisms is $n$ times the number of canonical orthomorphisms.

b) A Latin square is an $n \times n$ matrix containing $n$ distinct symbols in which each symbol occurs exactly once in each row and each column. For instance, $$\begin{matrix} 1 & 3 & 2 \\\\ 3 & 2 & 1 \\\\ 2 & 1 & 3 \end{matrix}$$ is a $3 \times 3$ Latin square. We can put it in a canonical form (which I call normalised) by permuting the columns so that the first row is in order, i.e. $$\begin{matrix} 1 & 2 & 3 \\\\ 3 & 1 & 2 \\\\ 2 & 3 & 1 \end{matrix}$$ Here the total number of Latin squares is $n!$ times the number of normalised Latin squares. There's another canonical form (which I call reduced) which has the first row and first column in order.

Answer 17

It is not an answer but it must have to do with the way our brain pick up a sample between a few ones. It must be the minimum of some function which can be implemented for real in the brain. I do not believe that there is a pure logical definition of "canonical" independently of the way our brain works. Experience : give me a number? What do you answer? 0 or 1 rarely $\pi$ or even 115674. The numbers 0 and 1 are canonical in some sense. Give me a basis of ${\bf R}^3$. The same holds $((1,0,0),(0,1,0),(0,0,1))$ I minimize the number of different digits and I pick them in my "basis" of canonical numbers. Well, interesting question.

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What is the canonical circle ? Ce circle in ${\bf R}^2$, centered at $(0,0)$ with radius $1$.

I know two numbers $0$ and $1$, the radius cannot be $0$ because it is not a (true) circle, so the radius is $1$, now the center could be $(0,0)$, $(0,1)$, $(1,0)$ or $(1,1)$ ? I prefer $(0,0)$, $0$ is simpler than $1$. How do you fit this example with category arguments?

BTW I have nothing against category theory, I like it. But I'm curious to see if this example fits general categorical arguments.

Answer 18

For me canonical often means derived from existing structure. Let's say I have an inner product space $S$ with some nice inner product ($\langle{}x,y\rangle{}$ for $x,y \in S$ ). Now let's say I need a norm on this space. I could pick an arbitrary one. There are many possible choices. I would consider the one induced by the inner product ( i.e. $\sqrt{\langle{}x,x\rangle{}}$ ) to be the canonical choice. Even if another was more useful for the problem at hand.

Answer 19

There are two kinds of meanings for terms used in mathematics:

• stipulated definitions, where the meaning of a term is only what is given by the symbolic definition, like 'field' or 'normal' as in a vector. The word may connote certain things, but whenever in doubt it's -only- the stipulated definition that matters.
• loose, informal terms that have only their native language meaning, like 'number' (by itself).

And some terms have multiple meanings, and so can have stipulated and loose versions.

There are a lot of uses of 'canonical' which are loose. I do know of one meaning that is mathematically defined. It comes out of proof theory, specifically rewrite systems.

In this area there are very formal, stipulated definitions of 'canonical': roughly (in non-stipulated terms) you specify a partial order on terms and then a set of rewrite rules from one term to another. Then if after a succession of rewrite rules has occurred to a term, and no more rules apply, then you are at a minimal element of the order, and that element is called 'canonical'.

Since all these loose ideas are formalized in the rewrite rule literature, the idea 'canonical' then has a formal definition.