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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?

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Bourbaki's books are very tough. I'd advise Rotman or Weibel's books for a softer intro into category theory. – shamovic Mar 8 '11 at 13:06
I haven't read Bourbaki, but if I had to guess, I'd say that the words "theory" and "structure" are borrowed from logic, not category theory. In first-order logic, a theory is essentially a language together with some axioms, and a structure is a set together with some functions and relations which satisfy the axioms of the theory. If you haven't studied logic yet, perhaps it would be profitable to read an introductory text before trying to tackle other bits of metamathematics. – Zhen Lin Mar 8 '11 at 15:40
@Zhen : I didn't mean they were borrowed from category theory, I asked if they correspond to the respective concepts, at least informally. – Alexei Averchenko Mar 8 '11 at 15:43
you shouldnt read boubaki as a textbook. its more of a reference or a paperweight. – yoyo Mar 8 '11 at 16:42
up vote 6 down vote accepted

I have seen a bit of Bourbaki's structure definition. The Bourbaki group defines structure roughly as a collection of sets with functions and relations on them. They take as an example a topological space which is a set together with some subset of its power set. Bourbaki's structures are anything which can be defined in this manner, like groups, etc.

Category theory studies classes of objects and their morphisms. It tries to classify constructions based on abstract properties of morphisms and diagrams. For instance, in category we have a description of a product without ever using "elements", and this definition of a product applies to all categories; whether or not a product actually exists is another issue. Sometimes it does, and sometimes it doesn't.

Category theory does not give us an obvious way of constructing familiar structures like groups in the first place, although there is interesting mathematics behind how far one can go just with the concept of a category.

I suggest forgetting Bourbaki altogether unless (a) you need a specific result or (b) you are interested in historical treatments. Since Bourbaki covered much, what you should choose instead depends on what you want to learn. If you are interested in Category Theory, keep looking at Mac Lane or try Awodey's book.

On the other hand, if you are interested in analyzing mathematics from its models and structure, try model theory. It's a branch of logic that studies the types of models that can occur for a given set of axioms and how those models differ, both from a first-order perspective (i.e. what can you distinguish just by first-order sentences) and from an outside perspective (are they isomorphic?). David Marker's book on model theory is a nice introduction.

If you are just interested in specific structures like groups, rings, etc. read some abstract algebra. You should have a good familiarity with abstract algebra, which will help you understand why category theory is so important. Depending on how much algebra you know, picking up an introductory book like Rotman's on Homological Algebra might interest you. You can also look at Weibel's book on Homological Algebra, which is my favourite, although it is more advanced.

Most graduate algebra textbooks introduce some really basic category theory and use it in algebra. Jacobson's Basic Algebra 2 is pretty nice and includes various diagram definitions.

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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.

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+1 for reminding the situation the Bourbaki group faced at their beginning (a parameter all too often forgotten when assessing the virtues of their enterprise). – Did Jul 24 '11 at 15:25
Yes, but Bourbaki's work continued through the period when category theory was being invented. Mac Lane [Concepts and categories in perspective, 1988] jokes, ‘perhaps my command of the French language was inadequate to the task of persuasion.’ More concretely, he [CWM, 1998, p.108] suggests that Bourbaki almost had the idea for what we would now call an adjoint functor, but their definition of ‘universal construction’ in the end just missed it. – Zhen Lin Jul 24 '11 at 16:48
Well, in terms of chronology, Eilenberg-MacLane, Serre, and Grothendieck had widely publicized a categorical viewpoint, of which Weil, Cartier, and the other Bourbaki people were certainly well aware. So it wasn't an issue of inventing anything, or catching on, or even competing notions, but only of whether "category" would be officially adopted in the formal Bourbaki texts. Since one of their avowed goals was to have everything be self-contained, they were in an awkward position. The notion of Grothendieck topology and such made clear that "category" was the winner... and they all saw that. – paul garrett Jul 24 '11 at 19:33

My take on this is that category theory is a well-defined mathematical theory with a small set of definitions and axioms specifically for describing and proving things for categories, with many instances of these theorems applicable throughout mathematics.

On the other hand, the Bourbaki school really provides an -approach- to all mathematics that is very formal. It's more of an approach, a general way of presentation than anything else. There's no Bourbaki 'theory' (collection of theorems). The Bourbaki approach is an attempt to present traditional mathematics in a particular manner.

So with the Bourbaki approach, you'll get a kind of culture of doing mathematics, but with category theory you'll get category specific theorems (which have their application to other specific fields of mathematics).

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