Re: Bourbaki and Categories

[Andre.Rodin@ens.fr]

When one defines, say, a group à la Borbaki, i.e. structurally, it usually goes
without saying that the defined structure is defined up to isomorphism. The
notion of isomorphism plays in this case the role similar to that of equality
in the (naive) arithmetic. In most structural constexts the distinction between
the "same" structure and isomorphic structures is mathematically trivial just
like the distinction between the "same" number and equal numbers. It may be not
specially discussed in this case exactly because it is very basic. The notion
of admissible map, say, that of group homomorphism, on the contrary, requires a
definition, which may be non-trivial.
The idea to do mathematics up to isomorphism is not Bourbaki's invention; it
goes back at least to Hilbert's "Grundlagen der Geometrie" of 1899. In this
sense the modern axiomatic method is structuralist. In his often-quoted letter
to Frege Hilbert explicitely says that a theory is "merely a framework" while
domains of their objects are multiple and transform into each other by
"univocal and reversible one-one transformations". Those who trace the history
of mathematical structuralism back to Hilbert are quite right, in my view.
I have in mind two issues related to CT, which suggest that CT goes in a
*different* direction - in spite of the fact that MacLane and many other
workers in CT had (and still have) structuralist motivations. The first is
Functorial Semantics, which brings a *category* of models, not just one model
up to isomorphism. From the structuralist viewpoint the presence of
non-isomorphic models (i.e. non-categoricity) is a shortcoming of a given
theory. From the perspective of Functorial Semantics it is a "natural" feature
of mathematical theories to be dealt with rather than to be remedied.
The second thing I have in mind is Sketch theory. I cannot see that Hilbert's
basic structuralist intuition applies in this case. In my understanding things
work in Sketch theory more like in Euclid. Think about circle and straight line
as a sketch of the theory of the first four books of Euclid's "Elements". I
would particularly appreciate, Michael, your comment on this point since I
learnt a lot of Sketch theory from your works.
I have also a comment about the idea to rewrite Bourbaki's "Elements" from a new
categorical viewpoint. Bourbaki took Euclid's "Elements" as a model for his
work just like did Hilbert writing his "Gundlagen". In my view, this is this
long-term Euclidean tradition of "working foundations", which is worth to be
saved and further developed, in particular in a categorical setting. I'm less
sure that Bourbaki's example should be followed in a more specific sense.
Bourbaki tries to cover too much - and doesn't try to distinguish between what
belongs to foundations and what doesn't. As a result the work is too long and
not particularly usefull for (early) beginners. I realise that today's
mathematics unlike mathematics of Euclid's time is vast, so it is more
difficult to present its basics in a concentrated form. But consider Hilbert's
"Grundlagen". It covers very little - actually near to nothing - of geometry of
its time. But at the same time it provided a very powerful model of how to do
mathematics in a new way, which greatly influenced mathematics education and
mathematical research in 20th century. In my view, Euclid's "Elements" and
Hilbert's "Grundlagen" are better examples to be followed.

best,
andrei


le 15/09/08 12:59, Michael Barr à barr@math.mcgill.ca a écrit :

> I don't know about this.  I took several courses in the late 1950s that
> seem to have been influenced by the structuralist ideas (certainly
> categories weren't mentioned; I never heard the word until Dave Harrison
> arrived in 1959) and each of them started by defining an appropriate
> notion of "admissible map".  I do not recall any special point being made
> of isomorphism and I think in general it was used for what we now call a
> bimorphism (1-1 and onto) even in cases, such as topological groups, when
> they were not isomorphisms.
>
> To be sure Bourbaki was not mentioned either, but this structuralist
> influence seemed strong.
>
> Michael
>