TITLE: Introduction to Bourbaki structures, and their relation to category theory
QUESTION [7 upvotes]: I just opened vol.1 of the Bourbaki treatise to take a look at how they define mathematical structure. I was amazed by its sheer complexity. Can you recommend an introductory text that wouldn't require as much effort to understand?
Also, a few related soft questions:
1) Is category theory more general than this theory?
2) In Bourbaki approach, theory corresponds to category, and structure corresponds to object, am I right?
3) I recently started reading MacLane's "Categories for a working mathematician", and category theory seems much simpler to understand. Is it an illusion due to less formal exposition, or is it really the case? If it's the latter, was this the reason why everyone adopted category theory in place of Bourbaki approach?

REPLY [8 votes]: When Bourbaki began, in the 1930s, there was no "category theory", for one thing. One of the issues the group was addressing was the lack of "modern" texts (not only in French), and various problems of rigor in some existing sources. With hindsight, their notion of "structure" was not a big success, and they themselves did not really use it in later volumes.
It wasn't a completely frivolous idea, insofar as one can observe the dynamics of interactions of "different" fundamental notions ("algebraic" and "topological", etc.) However, with hindsight, the Bourbaki group was naive about foundations and philosophy-of-mathematics, no matter their great strengths in mathematics per se. Even their attitude about analysis seems skewed. E.g., where's the PDE volume? :)
Leo Corry's book "Modern algebra and the rise of mathematical structures" includes a discussion of Bourbaki's "structures", and makes comparisons to both category theory and some other early competing notions.
But, among other conclusions, one can ignore Bourbaki's notion of "structure" in terms of the practice of mathematics, or even for reading Bourbaki (!).
Edit: also, I think we should distinguish "foundational" from "organizational" attempts/conceptions, although an approach can include both. It does not seem that set theory ever tried to provide organizational principles for mathematics, only foundational (and interesting questions of its own). In contrast, category theory has always been far more organizational than foundational (despite Lawvere's work, and many others more recently). To my perception, Bourbaki's "structures" had more an organizational than foundational intention, although, arguably, any "economy" of concepts should make foundational burdens lighter.