Re: Bourbaki and Categories

[Steve Lack <s.lack@uws.edu.au>]

Dear Andrei,

Sketches are not mathematical objects in their own right, in the same sense
that groups or spaces are. They are presentations (for theories), and have
status similar to other sorts of presentations (for groups, rings, etc.)

Of course that is in no way meant to suggest that they are not important and
worthy of study.

Regards,

Steve Lack.


On 16/09/08 11:09 PM, "Andre.Rodin@ens.fr" <Andre.Rodin@ens.fr> wrote:

>
> Of course, you are right about a point, I missed it! I must confess I didn't
> think about this example in precise terms. My claim is that sketch theory
> doesn't fit the structuralist (Bourbaki-Hilbertian) pattern. It hardly
> precisely fits the ancient Euclidean pattern either but there is a suggestive
> analogy, which concerns the idea that certain basic objects like point, line
> and circle *generate* the rest.
> A further claim is this: a specific reason *why* sketch theory doesn't fit the
> structuralist pattern is that in sketch theory (like in CT in general)
> isomorphisms don't have the same distinguished status.
>
> andrei
>
>
>
>
>
>> 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
>
>
>