I have seen a discussion about Bourbaki’s usage of categories before. So let me ask a different question: why did he not use universal algebra?

Well, universal algebra is not much older than category theory—however, it is older, and I think it would have fitted very well into Bourbaki’s general approach. Bourbaki even gave general notions of a “structure” and of homomorphisms in volume 0. However, in Topologie générale and Algèbre he started from scratch without applying this general concepts. I think these general concepts can be used to present all the different kinds of structures and morphisms used by Bourbaki in a very schematic way. Why did he not structure the definitions of particular structures by referring to the general notion? His definition of a structure is complicated and not widely used, but I do not think it would have been too complicated for applications. I cannot believe that Bourbaki was too dump to prove general theorems about their structures, even if universal algebra was not yet mature. Some examples:

He could have proven a general isomorphism problem using congruences and stating a special one for those cases where kernels are available. I do not know who did that first—but would it have been an absurd idea in the 40s? Or he could have introduced a general notion of free structures as done in universal algebra. Maltsev considered these concept quite generally (but it might have been in the 50s, I am not sure). For people who are used to nice categorical notions this approach to free structures (objects) might appear annoying (“why does he not simply tell me the category he is talking about?”), but it is still a very general and fruitful theory.

Are there pedagogical reasons? Well, have a look at Bourbaki’s definitions of sesquilinear forms or derivations: There he did not care about “let us start with an important special case”. He just gives nearly the most general definition he could imagine and he is able to work with it (I guess that most people have never seen such a definition of a derivation (it is defined involving six (!) graded modules and three bilinear forms), including graded derivations etc. as a special case). People who can do that should be able to give a general isomorphism theorem, too, I think.

In his volume 0 Théorie des ensembles he even introduces the concept of a structure in kinda philosophical way: He states that the information describing structures (the signature and formulas describing it) is not a formal, mathematical object, but a metamathematical, “philosophical” notion. Thus, in a strict interpretation, his general structures cannot be subject of mathematical statements (given in the formal language). However, this could have been easily circumvened—he could have defined formulas as formal objects with semantics with respect to some structure. But he did not—there is no model theory at all. 1941 Maltsev published group theoretic work applying the compactness theorem. Why did Bourbaki not apply such techniques? The preliminary works in logic had been there for a long time.

I do not understand it—Bourbaki’s general structural approach (and even his philosophy) calls for universal algebra, which seems like an obvious step to take (contrary to category theory, which may be regarded as a paradigm shift). But he just forgot what he told in volume 0 when he started with the real work.

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    $\begingroup$ It should be CW. Voted to close as it invites speculation. Why don't you ask this question one of the surviving members of Bourbaki and then report what you find out in your blog? $\endgroup$ – Mark Sapir May 24 '13 at 23:01
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    $\begingroup$ I am slightly exhausted on this subject for the moment, and also do not want to interfere with ongoing closure or not discussion, so just a quick comment (at least for now): Bourbaki or at least some of its early members, in particular Weil (but sometimes also roles changed depending on occassion), did not consider it as a goal to be as general as possible. An example in a letter 19th July 1946 Cartan asked Weil regaridng opinion on including an axiomatic theory of systems of generateurs (in this form due to him and mentioning an earlier version of van der Waerden). To which... $\endgroup$ – user9072 May 24 '13 at 23:33
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    $\begingroup$ Weil replied (July 30th 1946) that he is well aware that this is possible and the work of van der Waerden also mentions related work of Mac Lane and says this is all fine but he considers it as a mistake to include this and it were: "Bel exemple de fausse généralié!" [Nice example of false generality!] on the grounds that if one includes this one has still to translate the abstarct result to the special applications and little is gained by this, and he had some additional arguments more particular to the situation. Yet one might include this abstarct approach as an excercise. Now,.. $\endgroup$ – user9072 May 24 '13 at 23:41
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    $\begingroup$ ...this is a very rough description or rather paraphrase and an isolated instance. But still I hope it conveys some information that makes it a lot less suprising that things are not as general as possible. It was simply not considered as necessary or even desirable (at least by some). $\endgroup$ – user9072 May 24 '13 at 23:54
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    $\begingroup$ You should also recall that universal algebra in the 40's and 50's was quite a bit different than a mere two decades later. I am far from being an expert in the history of even specialized subjects as model theory or universal algebra, but the only concepts I can recall studying from that time are related to Birkhoff's HSP theorem or his and related studies in lattice theory. From my perspective, "not yet mature" is an understatement. Also, what were they using it for? My memory yields no French scholars from that time in the field. Gerhard "Or Anywhere Close To Logic" Paseman, 2013.05.24 $\endgroup$ – Gerhard Paseman May 25 '13 at 6:56

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