The Association of Collaborators of Nicolas Bourbaki (l'Association des collaborateurs de Nicolas Bourbaki) is a collective of mathematicians based in France, who in the 20th century set a standard for a structural and abstract approach to exposition of pure mathematics. They published a series “Elements of mathematics” of 10 multi-volume monographs under the pseudonym ‘Nicolas Bourbaki’, consolidating known results. They also held —and still hold— a largely independent seminar on contemporary research.
The members of the group at any one time are required to be under 50 years of age, which is considered the only fixed rule of the group^{1}; the membership is taken under acceptance of an invitation by existing members. For a bit of history can be found at en.wikipedia, fr.wikipedia, britannica and references therein.
According to Jean Dieudonné, the Bourbaki series of expositions is dedicated to “dead”, i.e. stable mathematics which is not likely to change soon and which has wider importance, rather than being of a specialized character. Most successful was the exposition of the chapters on Lie theory, especially concerning Lie algebras, whose style is later followed in much of later literature in the subject. See more in (Dieudonné).
Bourbaki has been blamed for following too formal an approach. Indeed, the books are void of much motivation and application, apart from a few of introductions and few chapters on history.
Apart from the Bourbaki volumes, there is also a lively Bourbaki seminar which takes place on certain Saturdays in Paris, is open to public, has non-Bourbaki members as invited speakers who present and discuss in advance chosen topics of recent development in mathematics; the expositions are subsequently published.
A central point of view of Bourbaki is the emphasis of mathematics as the study of structure, as in the approach developed by Göttingen mathematicians in the 1920s and 1930s, such as Emmy Noether and Emil Artin, and written up by van der Waerden in Moderne Algebra (see Corry). This approach was very influential in the mainstream mathematics of the second half of the 20th century.
However, Bourbaki did not embrace category theory, which may be thought of as being the essence of that structural approach, though some of the universal properties treated in category theory in fact first appeared in early editions of Bourbaki. Instead, Bourbaki proposed its own formalization of the notion of “structure” (in the book Theory of Sets), which however neither caught on nor does it seem to have been taken very seriously by the group itself. As documented by Corry, the Elements pays only the slightest lip service to the Bourbaki formal notion of structure in its vast development (in the books after Theory of Sets when they are considered in their intended logical order).
Discussion of this point can be found in (CTList). There Colin McLarty writes:
Bourbaki’s first publication was
Bourbaki, N. [1939]: Théorie des ensembles, Fascicules de résultats, 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]: Théorie 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 Dieudonné, 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. (McLarty, Category Theory List 2012)
Accounts of the history of the Bourbaki project include the following
Bourbaki’s biography at math tutor: the pre-war years, the after-war years
Amir D. Aczel, The artist and the mathematician: the story of Nicolas Bourbaki, the genius mathematician who never existed, Thunder’s Mouth Press, New York, 2006, viii+239 pp. MR2008h:01020, SIAM review
The Bourbaki archives contain many draughts (and discussions thereof) of Bourbaki’s texts
Discussion of Bourbaki’s notion of structure and the relation (or not) to category theory includes the following:
Jean-Michel Kantor, Bourbaki’s Structures and Structuralism, The Mathematical Intelligencer 33:1 (2011), 1, doi
Leo Corry, Chapter 7 Nicolas Bourbaki: Theory of structures (pdf)