Return to Question

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.

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.

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 bilinear 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 using six (!) graded modules), 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.

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.

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 bilinear 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, 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.

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 bilinear 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 using six (!) graded modules), 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.