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In this question: What is the definition of "canonical" ? , people gave interesting "philosophical" takes on what the word "canonical" means. Moreover I percieved an underlying opinion that there was no formal mathematical definition.

Whilst looking for something else entirely, I just ran into Bill Messing's post


on the FOM (Foundations of Mathematics) mailing list. I'll just quote the last paragraph:

"It is my impression that there is very little FOM discussion of either Hilbert's epsilon symbol or of Bourbaki formulation of set theory. In particular the chapitre IV Structures of Bourbaki. For reasons, altogether mysterious to me, the second edition (1970) of this book supressed the appendix of the first edition (1958). This appendix gave what is, as far as I know, the only rigorous mathematical discussion of the definition of the word "canonical". Given the fact that Chevalley was, early in his career, a close friend of Herbrand and also very interested in logic, I have guessed that it was Chevalley who was the author of this appendix. But I have never asked any of the current or past members of Bourbaki whom I know whether this is correct."

It's a 4-day weekend here in the UK and I'm very unlikely to get to a library to find out what this suppressed appendix says. Wouldn't surprise me if someone could find this appendix on the web somewhere though! Is there really a mathematical definition of "canonical"??

NOTE: if anyone has more "philosophical" definitions of the word, they can put them in the other thread. I am hoping for something different here.

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@Qiaochu: I have in my hand a paper by Nick Katz ("p-adic properties of modular schemes and modular forms") where he says that a certain exact sequence coming from the theory of elliptic curves has a "canonical, but not functorial, splitting" (page Ka-95, a.k.a. p163 of the book). –  Kevin Buzzard Apr 2 '10 at 10:34
In the spirit of the old nonsense about a tree which falls in the forest when nobody is around to hear the noise, if a mathematical concept is defined in a place which is known to almost nobody then the definition may as well not exist. Or in the words of a US Supreme Court case struggling with another widely used undefined word, as long as we know it when we see it then probably there's no need for a rigorous definition (much like there's no need for rigorous foundations of category theory as long as it is being used in an essentially linguistic or "plug example into machine" manner). –  BCnrd Apr 2 '10 at 15:15
Manjul Bhargava got a big laugh in one of his early talks on his higher composition laws. In the difficult degree five case, someone asked if the number field entity Manjul constructed had a certain desirable property, and he replied "No, but it's unique." en.wikipedia.org/wiki/Manjul_Bhargava –  Will Jagy Apr 2 '10 at 15:15
@Dmitri: When writing my comments I had in mind what you say, but we could always go a step further and pass to the discrete subcategory (with only identity morphisms) to make "everything" canonical. :) So I thought the main interesting feature of those little examples I mentioned was that one can make natural useful constructions whose functoriality is very restricted compared with where we begin (surjections of appropriate sort for the unipotent radical and orthogonal complement examples, etc.) –  BCnrd Apr 2 '10 at 18:52
Here's a nice little example. I'd say that the center of a group a canonical construction, but it does not prolong to a functor on the category of groups. –  JBorger Apr 4 '10 at 11:24
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2 Answers

up vote 12 down vote accepted

Although the Bourbaki formulation of set theory is very seldom used in foundations, the existence of a definable Hilbert $\varepsilon$ operator has been well studied by set theorists but under a different name. The hypothesis that there is a definable well-ordering of the universe of sets is denoted V = OD (or V = HOD); this hypothesis is equivalent to the existence of a definable Hilbert $\varepsilon$ operator.

More precisely, an ordinal definable set is a set $x$ which is the unique solution to a formula $\phi(x,\alpha)$ where $\alpha$ is an ordinal parameter. Using the reflection principle and syntactic tricks, one can show that there is a single formula $\theta(x,\alpha)$ such that for every ordinal $\alpha$ there is a unique $x$ satisfying $\theta(x,\alpha)$ and every ordinal definable set is the unique solution of $\theta(x,\alpha)$ for some ordinal $\alpha$. Therefore, the (proper class) function $T$ defined by $T(\alpha) = x$ iff $\theta(x,\alpha)$ enumerates all ordinal definable sets.

The axiom V = OD is the sentence $\forall x \exists \alpha \theta(x,\alpha)$. If this statement is true, then given any formula $\phi(x,y,z,\ldots)$, one can define a Hilbert $\varepsilon$ operator $\varepsilon x \phi(x,y,z,\ldots)$ to be $T(\alpha)$ where $\alpha$ is the first ordinal $\alpha$ such that $\phi(T(\alpha),y,z,\ldots)$ (when there is one).

The statement V = OD is independent of ZFC. It implies the axiom of choice, but the axiom of choice does not imply V = OD; V = OD is implied by the axiom of constructibility V = L.

When I wrote the above (which is actually a reply to Messing) I was expecting that Bourbaki would define canonical in terms of their $\tau$ operator (Bourbaki's $\varepsilon$ operator). However, I was happily surprised when reading the 'état 9' that Thomas Sauvaget found, they make the correct observation that $\varepsilon$ operators do not generally give canonical objects.

A term is said to be 'canonically associated' to structures of a given species if (1) it makes no mention of objects other than 'constants' associated to such structures and (2) it is invariant under transport of structure. Thus, in the species of two element fields the terms 0 and 1 are canonically associated to the field F, but $\varepsilon x(x \in F)$ is not since there is no reason to believe that it is invariant under transport of structures. They also remark that $\varepsilon x(x \in F)$ is actually invariant under automorphisms, so the weaker requirement of invariance under automorphisms does not suffice for being canonical.

To translate 'canonically associated' in modern terms:

1) This condition amounts to saying that the 'term' is definable without parameters, without any choices involved. (Note that the language is not necessarily first-order.)

2) This amounts to 'functoriality' (in the loose sense) of the term over the core groupoid of the concrete category associated to the given species of structures.

So this seems to capture most of the points brought up in the answers to the earlier question.

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Francois: you are telling me precisely the kind of mathematics that I had hoped this question would inspire. I am currently way behind though. I am happy with V=L. Let me try and translate what you say in your second para above. You say "let's well-order everything with <. Now given a non-empty bunch of sets we can take the 'smallest' one (wrt this ordering). Now if F is an arbitrary field of order 2, its 'smallest' element is automorphism-invariant (as Aut(F)=1) but not isomorphism-invariant (as if F and G are fields of order 2, the smallest elt in one might be 0 but in the other might be 1)" –  Kevin Buzzard Apr 2 '10 at 15:30
So that's an example of something not canonical. But if I have enough mathematics to isolate the 0 and 1 of a field, clearly these are canonical---because they're isomorphism-invariant? But all this seems very far away from the statement (which I truly believe) that the isomorphisms of local class field theory (sending a uniformiser to a geometric Frobenius) are canonical. Am I now using canonical in a different way? Can one get from terms and functoriality to local class field theory isomorphisms? –  Kevin Buzzard Apr 2 '10 at 15:30
Kevin, this is correct. The point can be summarized as follows: V = OD gives a canonical way of making choices, but not a way of making canonical choices. –  François G. Dorais Apr 2 '10 at 15:32
This is not my specialty so I can't say for sure, but I think sending a uniformizer to the geometric Frobenius is canonical and so is the inverse convention. The Bourbaki language of species is not first-order, so you can talk about a variety of higher-order objects (e.g. idèles and adèles would certainly be canonical) so the fact that there are small variations shouldn't affect things very much. –  François G. Dorais Apr 2 '10 at 15:41
@Francois: class field theory isn't hard to summarise. Let me try and abstractify the number theory away. I have a fixed field Q_p. If F is any finite extension of Q_p then I can associate two topological groups canonically to F: call them Wab(F) [the abelianisation of the Weil group associated to F] and M(F) [the multiplicative group F^*]. Now Wab() and M() can be extended to covariant functors from the cat of finite extensions of Q_p (with the morphisms being inclusions) to the cat of topological groups, and they can also be extended to contravariant functors between these categories! –  Kevin Buzzard Apr 2 '10 at 16:00
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There are scanned notes in french that were used for the initial text of Théorie des Ensembles on the Bourkaki Archives website.

In particular there are indeed notes by Chevalley named Livre I. Théorie des ensembles Chap. IV (état 7 ?) Structures (53 p.) which seem at first glance to define "canonique" in the broader context of "transport de structures, idendifications" (see exemple 1 at the bottom of page 19 of that file).

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That's a very interesting website! I'm not convinced that the reference you give is what Messing is referring to though. It seems to me that they are just giving some standard examples of canonical isomorphisms (e.g. "the integers" (however they have defined them) are canonically isomorphic to the subset of the rationals consisting of things which are integers...) –  Kevin Buzzard Apr 2 '10 at 10:40
You're right. I had another look and I think the relevant file is the last one (état 9), which does have an appendix, as Messing mnetions, in which at page 37 and 38 we find: "un terme U est canoniquement associé à la structure générique $(s_1,...,s_p)$" to be defined as "U ne contient aucune lettre autre que les constantes de $T_\Sigma$ et est transportable relativement à $\Sigma$". It is also given the alternative name "intrinsèque pour $(s_1,...,s_p)$". –  Thomas Sauvaget Apr 2 '10 at 11:06
You could well be right about \'etat 9. At first glance---I can't make head nor tail of it! I wonder what it says! –  Kevin Buzzard Apr 2 '10 at 11:27
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