Re: Bourbaki & category theory

[Colin McLarty <colin.mclarty@case.edu>]

On Wed, May 23, 2012 at 7:36 AM, George Janelidze <janelg@telkomsa.net> wrote:
> I am ready to take back my criticism and apologize, if the "longer sentence"
> is correct. But is it?

Yes, it is.  I am an expert on Bourbaki history.  See my articles on
Chevalley, Dieudonné, and Weil in the New Dictionary of Scientific
Biography. As Eduardo says, it is a historical question.  Here are the
historical facts

Bourbaki's first publication was

Bourbaki, N. [1939]: Th{\'e}orie des ensembles, Fascicules de
r{\'e}sultats, Paris:
Hermann, Paris.

It is very sketchy on "structures," and uses no notion of mapping
between structures except isomorphisms. Their actual theory of
structures first appeared in

Bourbaki, N. [1957]: Th{\'e}orie des ensembles}, Chapter 4, Paris: Hermann.

That theory was a rear-guard action meant to give an alternative to
category theory.  As i mentioned before, Weil was citing the
categorical idea, and thinking about finding an in-house alternative
to it, already in 1951.  By 1957 Grothendieck, and Cartier, and
Chevalley, probably Dieudonne, and others, all saw that category
theory was more agile than these structure, simpler, and more to the
point, plus it had a natural "higher order" aspect in the theory of
functors which was actually more useful in practice than categories
alone.

Cartier has justly said it would have been a huge job to formulate all
Bourbaki's ideas in terms of categories and functors.  It would have
called for a lot of ideas which were only invented in the coming
years.

It was relatively easy to give Bourbaki's theory of structures --
because it never really worked at all even for Bourbaki's purposes (as
Corry documents in detail).  Naturally it is easier to give an
unusable theory of structures than to work out the ways categories and
functors would actually be used.

best, Colin


>
> I am certainly not an expert in Bourbaki history, and, as far as I remember,
> they say no word about morphisms in the historical part of "Theory of Sets"
> and give no references on categories. But I think they "always" believed
> that structures determine isomorphisms but not morphisms, and I don't think
> they changed their mind between 1951 and 1957.
>
> When I say "they" I mean "those of them who made main decisions about the
> Bourbaki tractate". Because I hope (!) that not all of them were happy that
> categories are not even defined in the tractate.
>
> In my previous message I wrote "Removing Bourbaki's formalism..." but in
> fact that "formalism" is (not nice but) serious, in the sense that it takes
> us further away from abstract categories.
>
> Anyway, we need to know, if it is still possible, how exactly did Bourbaki
> definition of morphism(s) came up.
>
> George
>
>

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