Dear Colleagues, I don't think it is good to say that "Bourbaki had a notion of isomorphism but no general notion of morphism", even in a brief message! Bourbaki introduces morphisms in subsection 1 of section 2 of Chapter IV of "Theory of Sets". Removing Bourbaki's formalism, the definition can be stated as follows: Let S be a class of mathematical structures of a given type, and let us assume for simplicity that these structures have single underlying sets (Bourbaki also makes this assumption, also just for simplicity). An element of S is therefore a pair (x,s), where x is a set and s a structure on x. By a map f : (x,s) --> (y,t) we shall mean an arbitrary map f from x to y. But, just as in category theory, a map (x,s) --> (y,t) should "remember" (x,s) and (y,t). A class M of such maps is said to be a class of morphisms if it satisfies the following conditions: (i) If f : (x,s) --> (y,t) and g : (y,t) --> (z,u) are in M, then so is gf : (x,s) --> (z,u); (ii) let f : (x,s) --> (y,t) be a map that is a bijection x --> y, and let g : (y,t) --> (x,s) be its inverse, then f is an isomorphism if and only if both f and g are in M. Here Bourbaki uses the notion of isomorphism that he (they) defined before and that is completely determined by S. That is, according to Bourbaki, as soon as I know the structures I also know their isomorphisms - but I still might have flexibility in defining morphisms. How flexible it is? Well, condition (i) says that the class of morphisms must be closed under composition, and condition (ii) says that morphisms must nicely agree with isomorphisms. The readers not familiar with category theory might ask, why is it so? Why should we agree that isomorphisms are "more determined" than morphisms? And one needs very little category theory to answer this: Bourbaki structures are defined as elements of sets that appear in "scales" build using finite cartesian products and power sets (in which we see a "germ" of Pare theorem about toposes: no colimits!), and to define morphisms we need functoriality of scales that we don't have, since we might want to use the contravariant power set functor, as e.g. for topological spaces. And this problem disappears of course if we restrict ourselves to bijections. (In fact even without power sets there is a problem that disappears for bijections, but never mind). The citation from Weil might mean that he misunderstood (or almost understood) exactly this... (I mean, not structures, but parts of scales behave covariantly or contravariantly! - But apologizing to Weil, I must say that I have not read the full text - so maybe it is I who misunderstood him... for instance Mac Lanes idea could be that in many cases there is the "best" notion of morphism, which is correct of course). Talking about Bourbaki and categories, how can we not mention Charles Ehresmann who was far ahead in understanding many aspects of category theory, and actually used the category of sets and bijections to approach the general concept of mathematical structure?! (It is interesting that his concept of a mathematical structure is briefly mentioned in "Introduction to the theory of categories and functors", a book written by I. Bucur and A. Deleanu). Surely Andree Ehresmann can tell us many interesting things about his ideas and the reaction of the rest of Bourbaki on them. But let me return to Bourbaki in general. Seeing that there is certain flexibility in choosing morphisms is great, but then not seeing that the morphisms do not have to be maps of sets at all is absolutely crucial. So, yes, category theory theory was definitely invented by Mac Lane and Eilenberg and not by Bourbaki. George Janelidze P.S. Also, as we all know, Einenberg and Mac Lane wrote "General theory of natural equivalences" long before Bourbaki wrote their Chapter IV, which itself was long before 1970. -------------------------------------------------- From: "Colin McLarty" <colin.mclarty@case.edu> Sent: Tuesday, May 22, 2012 7:25 PM To: "Staffan Angere" <Staffan.Angere@fil.lu.se> Cc: <categories@mta.ca> Subject: categories: Re: Bourbaki & category theory

Prior to encountering category theory, Bourbaki had a notion of isomorphism but no general notion of morphism. See this letter from. Andre Weil to Claude Chevalley, Oct. 15, 1951:

\begin{quotation} As you know, my honorable colleague Mac~Lane maintains every notion of structure necessarily brings with it a notion of homomorphism, which consists of indicating, for each of the data that make up the structure, which ones behave covariantly and which contravariantly [\dots] what do you think we can gain from this kind of consideration? (quoted in Corry ~\cite[p. 380] Modern Algebra and the Rise of Mathematical Structures}, Basel: Birkh{\"a}user 1996.\end{quotation}

On Mon, May 21, 2012 at 6:49 PM, Staffan Angere <Staffan.Angere@fil.lu.se> wrote:

...

Dear categorists,

and also, hello everyone, since this is my first post here! I'm wondering about the connection of Bourbaki to category theory. The copy of "Theory of Sets" that I have says it's written in 1970. Yet, Dieudonné famously saiid that the theory of functors subsumed Bourbaki's theory of structures... and, also, Bourbaki's theory of structures is very clearly a theory of a type of concrete categories. On the other hand, I've seen claims that the categorists' use of "morphism" comes from Bourbaki. So who was first? Does anyone here know when Bourbaki's theory of structures was really conceived? I guess this might be self-evident to anyone born during the 1st half of the 20th century, but it has turned out to be really hard to find out for me.

Thanks in advance, staffan

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On 22/05/12 14:49, Eduardo J. Dubuc wrote:

The reason why category theory "is what it is" is that

it is the language that allows to define the notion of universal property in its right generality.

The notion of universal property first appears in Bourbaki, which decided not to use the language of categories to formulate it, on spite of the advice of Grothendieck.

Concerning the remark "on spite of the advice of Grothendieck" (that I concluded following some readings I do not remember where, and that if I remember correctly, Grothendieck wanted to rewrite the whole project (before it was published) using Category Theory, and presented a proposal that after some consideration was turned off by Bourbaki) I received a private mail that can be of interest to many and that I copy and paste below: =================================================================== Pierre Cartier, Chritian Houzel, Andrej Rodin and Ralf Krömer wrote about that issue. You will find the source material, i.e. the early Bourbaki archives (1934-1954), which show various early attempts at defining "structure", online at http://mathdoc.emath.fr/archives-bourbaki/ A bibliography of sources on Bourbaki is at http://poincare.univ-nancy2.fr/Actu/?contentId=9473 One may not confuse any edition of the first volume of the Elements with Bourbaki relatively evolving thoughts on this matter. There is a lot to be learnt from reading the sources. Please notice that I replied to you personally and not to the whole list. Be kind to a shy person. Best regards, LB ==================================================================== e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

The topic "Bourbaki and category theory" has several interesting aspects, and I shall write at least one more message about that. At the moment, under the permission of Francis Borceux, I would like to show one of his messages to me written a few days ago: "...I had a look at Nicolas Bourbaki Elements de mathematiques Theorie des ensembles Chapitre 4 Second edition Hermann 1966 just to refresh my mind. The first edition of this book had been published in 1957. First of all an obvious remark. The work of Bourbaki is by no means a "one shot work". It took many decades to produce it. Many books and chapters have had successive editions, sometimes very different from the previous ones. A new edition was generally reflecting all the progresses and changes of point of view made since the previous edition. And if generally a new edition is "Revised and augmented", the second edition of the book mentioned above is at once presented as "Revised and reduced...". I am sure that many people would be able to comment infinitely on this peculiar aspect. The point of Bourbaki in Section 1 of this Chapter 4 is to define what a mathematical structure is ("une espece de structure"). This is done in a very general setting, which includes most mathematical structures you can think of, whatever their nature: algebraic, topological, ordered, categorical, and so on. It is in this first section that the notion of "isomorphism" is investigated. Section 2 of Chapter 4 is devoted to the notion of "morphism" for such a mathematical structure (this Section is called "Morphismes et structures derivees" ... thus even the title of the section refers explicitly to the notion of "morphism"). The notions of "initial structure" and "final structure", so popular in categorical topology, are at once investigated. The notion of product is presented as a special case of an initial structure. Section 3 is then devoted to the general "universal problem" when comparing two mathematical structures. The "existence theorem" is stated and proved in Subsection 3.2 in terms of the existence of products and a "solution set condition". So what? Well, category theory intends to study objects and morphisms, without any reference to a mathematical structure which would have given rise to them. And this proved to be the very natural and efficient context where to introduce the notions of limit and adjoint functors and prove the basic theorems about them. But indeed, the "adjoint functor theorem" is essentially present in Bourbaki, even if the abstract notions of "category" and "functor" never appear.? The theorem is proved in the case of "two mathematical structures" in the very general sense defined by Bourbaki. The great merit of Peter Freyd has been to put the "universal problem for two structures" in the elegant context of categories and adjoint functors. And to have given a corresponding elegant proof of the "adjoint functor theorem". This has made the question fully transparent, in opposition to the heavy technicalities found in Bourbaki. Now was Peter Freyd aware of the result of Bourbaki? Probably, since in those days Mac Lane and Eilenberg had regular contacts with the Bourbaki group. But there is no shame at all -- just merit -- to generalize an existing result, especially to put it in its "right context". But why to care about these questions of priorities? Everybody knows the Fermat theorem ... but did he really prove it? As a matter of comparison, also the "nine lemma" and the "snake lemma" did exist before the invention of abelian categories. But abelian categories provided a beautiful and natural context where to study these lemmas. And of course, these lemmas have been further investigated in much more general contexts than just abelian categories. Like for adjoint functors, further studied in enriched, bi, 2, pseudo or lax contexts. To whom should we give credit for such results? To the author of the very first result of that kind? To the author of the more general result? To the author of the result which "you" consider as most "natural". Really, I am not interested in argueing on this. Did you already count the number of "Pythagoras theorems" in mathematics ... or the number of "Galois theorems"? Now, all right. As a "has been" category theorist, I consider abstract categories as the most natural setting for studying the adjoint functor theorem. But I am also aware that rapidly category theorists leave the context of "abstract" categories for more specific "mathematical structures" in order to prove more precise theorems. They study theories giving rise to algebraic categories, accessible categories, topological categories, classifying toposes, and so on. All these theories (Lawvere, sketches, coherent theories, ...) fall under the scope of Bourbaki's "structures". Thus Bourbaki did prove his "universal mapping theorem" in a general setting which includes (probably) all concrete mathematical examples that you can find in categorical books as applications of the more general Freyd adjoint functor theorem. But nevertheless, as far as universal problems are concerned, I consider Freyd's approach as much more elegant..." End of message of Francis Borceux copied by George Janelidze [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Let me add to the remarks on the extent to which Bourbaki did -- or did not -- envisage categories or category theory as an appropriate vehicle for conveying universal constructs in their expositions of other mathematics, by mentioning in this context the late Serge Lang's now half-century-old report, originally prepared "for Bourbaki's internal consumption", on the cohomology of groups: {Rapport sur la Cohomologie des Groupes}, W.A. Benjamin, NY, 1966. On page 97 of the Benjamin publication, Serge seeks to "definir la notion de catégorie multilinéaire", notion which his PREFACE indicates is "due to Cartier" and which figures not only in these 1959-era notes but also in the first-year graduate algebra course he gave at Columbia during the 1958-1959 academic year. (It was my first year at Columbia, and I remember that course well -- it gave me not only my first exposure too Serge Lang, and to categories, but to Saul Lubkin and Peter Freyd as well, each of whom Serge invited to give a "guest lecture" for a day (it was the era of the "race for the best embedding theorem":-) ). Whatever else may have been the case then or earlier as regards Bourbaki and categories, by 1959 even multilinear categories were being pressed upon Bourbaki in certain quarters, by certain proponents, without, alas, much perceptible effect. Cheers, -- Fred ------ Original Message ------ Received: Thu, 24 May 2012 08:28:11 PM EDT From: rlk@knighten.org To: "categories@mta.ca" <categories@mta.ca> Subject: categories: Re: Bourbaki & category theory [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Eduardo J. Dubuc writes:

. . . The notion of universal property first appears in Bourbaki, . . .

The notion and name (well "universal mappinga") first appears in: @article {MR0025152, AUTHOR = {Samuel, P.}, TITLE = {On universal mappings and free topological groups}, JOURNAL = {Bull. Amer. Math. Soc.}, FJOURNAL = {Bulletin of the American Mathematical Society}, VOLUME = {54}, YEAR = {1948}, PAGES = {591--598}, ISSN = {0002-9904}, MRCLASS = {56.0X}, MRNUMBER = {0025152 (9,605f)}, MRREVIEWER = {S. Mac Lane}, } Of course Samuel was a member of Bourbaki at the time but this seems to be the influence of Samuel on Bourbaki rather than the other way around. Note that Mac Lane reviewed this and he was quick to use both the notion and the terminology in his own writings. -- Bob -- Robert L. Knighten 541-296-4528 RLK@knighten.org [For admin and other information see: http://www.mta.ca/~cat-dist/ ]