Re: Bourbaki & category theory

[maxosin@berkeley.edu]

Dear Staffan,

Bourbaki has laid the set theory-based foundations for the Elements of
Mathematics in Théorie des ensembles Résumé in late 1930s,  and when
category theory has matured as an alternative foundation for mathematics
they decided that it was too late for them to change horses in the
midstream. Their time was limited and they had to scale down the original
set-theoretical project several times anyway. Refer to A. Borel's article
"Twenty-Five Years with Nicolas Bourbaki, (1949 – 1973)" for more  detail.

Mac Lane hints that there may have been other reasons as well:

"Categorical ideas might well have fitted in with the general program of
Nicolas Bourbaki for the systematic presentation of mathematics. However,
his first volume on the notion of mathematical structure was prepared in
1939 before the advent of categories. It chanced to use instead an
elaborate notion of an dchelle de structure which has proved too complex
to be useful. Apparently as a result, Bourbaki never took to category
theory. At one time, in 1954, I was invited to attend one of the private
meetings of Bourbaki, perhaps in the expectation that I might advocate
such matters. However, my facility in the French language was not
sufficient to categorize Bourbaki. Perhaps the explanation for his
resistance is the hard fact that categories were not made in France. Even
Eilenberg's later
membership in Bourbaki did not serve to overcome Bourbaki's
disinclination. It may be that the circulation of new ideas is not always
unhindered."
(Applied Categorical Structures, Vol. 4, No. 2-3 (1996), 129-136)

Comparing my copies of old editions of Bourbaki with latest editions, I
see that he nevertheless sneaks categorical language here and there
without, however, calling it explicitly category theory.

Max

> On 21/05/12 19:49, Staffan Angere wrote:
>> Dear categorists,
>>
>> and also, hello everyone, since this is my first post here! I'm
>> wondering about the connection of Bourbaki to category theory. The copy
>> of "Theory of Sets" that I have says it's written in 1970. Yet,
>> Dieudonné famously saiid that the theory of functors subsumed Bourbaki's
>> theory of structures... and, also, Bourbaki's theory of structures is
>> very clearly a theory of a type of concrete categories. On the other
>> hand, I've seen claims  that the categorists' use of "morphism" comes
>> from Bourbaki. So who was first? Does anyone here know when Bourbaki's
>> theory of structures was really conceived? I guess this might be
>> self-evident to anyone born during the 1st half of the 20th century, but
>> it has turned out to be really hard to find out for me.
>>
>> Thanks in advance,
>> staffan




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