Re: Bourbaki and Categories

[Michael Barr <barr@math.mcgill.ca>]

I don't know what to say about the suggestion that a circle and a line 
make a sketch of which Euclidean plane geometry is a model.  I would think 
you would need a point too, since intersections are crucial.  Maybe 
complex projective geometry since then two lines intersect in one point 
(unless they coincide), a line and a circle in two (unless they are 
tangent or equal) and every pair of circles in four (ditto).  Maybe the 
exceptions could be handled in some sketch.  At any rate, it wold e 
interesting to try to sketch this in detail.  At any rate, I never thought 
about this before.

Michael

On Tue, 16 Sep 2008, Andre.Rodin@ens.fr wrote:

>
> 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
>>
>