Bourbaki & category theory

Bourbaki & category theory

[Staffan Angere]

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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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Re: Bourbaki & category theory

[Colin McLarty]

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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Re: Bourbaki & category theory

[Eduardo J. Dubuc]

On 21/05/12 19:49, Staffan Angere 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
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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.

e.d


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Re: Bourbaki & category theory

[Ross Street]

On 23/05/2012, at 3:25 AM, Colin McLarty wrote:

> Prior to encountering category theory, Bourbaki had a notion of
> isomorphism but no general notion of morphism.  

Perhaps relevant (categories vs Bourbaki) is my memory that Sammy Eilenberg 
told me he and Chevalley invented the words injective, surjective and
bijective (as pertaining to functions) while strolling along a beach.

Also I heard Dieudonné admit that Bourbaki would have profited at least from the 
categorical notion of duality. 

==Ross

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Re: Bourbaki & category theory

[George Janelidze]

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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Re: Bourbaki & category theory

[Colin McLarty]

On Tue, May 22, 2012 at 5:51 PM, George Janelidze <janelg@telkomsa.net> wrote:
> 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!

It would not be good -- unless it was part of a longer sentence.

I wrote "Prior to encountering category theory, Bourbaki had a notion
of isomorphism but no general notion of morphism."  The Bourbaki
passage you quote was first published in 1957, at least 6 years after
Bourbaki encountered category theory as shown by the letter from Weil
that i quoted.

best, Colin


>
> 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.
>

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Re: Bourbaki & category theory

[George Janelidze]

I am ready to take back my criticism and apologize, if the "longer sentence" 
is correct. But is it?

I am certainly not an expert in Bourbaki history, and, as far as I remember, 
they say no word about morphisms in the historical part of "Theory of Sets" 
and give no references on categories. But I think they "always" believed 
that structures determine isomorphisms but not morphisms, and I don't think 
they changed their mind between 1951 and 1957.

When I say "they" I mean "those of them who made main decisions about the 
Bourbaki tractate". Because I hope (!) that not all of them were happy that 
categories are not even defined in the tractate.

In my previous message I wrote "Removing Bourbaki's formalism..." but in 
fact that "formalism" is (not nice but) serious, in the sense that it takes 
us further away from abstract categories.

Anyway, we need to know, if it is still possible, how exactly did Bourbaki 
definition of morphism(s) came up.

George

--------------------------------------------------
From: "Colin McLarty" <colin.mclarty@case.edu>
Sent: Wednesday, May 23, 2012 1:06 AM
To: "George Janelidze" <janelg@telkomsa.net>
Cc: <categories@mta.ca>
Subject: Re: categories: Re: Bourbaki & category theory

> On Tue, May 22, 2012 at 5:51 PM, George Janelidze <janelg@telkomsa.net> 
> wrote:
>> 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!
>
> It would not be good -- unless it was part of a longer sentence.
>
> I wrote "Prior to encountering category theory, Bourbaki had a notion
> of isomorphism but no general notion of morphism."  The Bourbaki
> passage you quote was first published in 1957, at least 6 years after
> Bourbaki encountered category theory as shown by the letter from Weil
> that i quoted.
>
> best, Colin
>
>

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Re: Bourbaki & category theory

[maxosin]

Dear Staffan,

Bourbaki has laid the set theory-based foundations for the Elements of
Mathematics in Théorie des ensembles Résumé in late 1930s,  and when
category theory has matured as an alternative foundation for mathematics
they decided that it was too late for them to change horses in the
midstream. Their time was limited and they had to scale down the original
set-theoretical project several times anyway. Refer to A. Borel's article
"Twenty-Five Years with Nicolas Bourbaki, (1949 – 1973)" for more  detail.

Mac Lane hints that there may have been other reasons as well:

"Categorical ideas might well have fitted in with the general program of
Nicolas Bourbaki for the systematic presentation of mathematics. However,
his first volume on the notion of mathematical structure was prepared in
1939 before the advent of categories. It chanced to use instead an
elaborate notion of an dchelle de structure which has proved too complex
to be useful. Apparently as a result, Bourbaki never took to category
theory. At one time, in 1954, I was invited to attend one of the private
meetings of Bourbaki, perhaps in the expectation that I might advocate
such matters. However, my facility in the French language was not
sufficient to categorize Bourbaki. Perhaps the explanation for his
resistance is the hard fact that categories were not made in France. Even
Eilenberg's later
membership in Bourbaki did not serve to overcome Bourbaki's
disinclination. It may be that the circulation of new ideas is not always
unhindered."
(Applied Categorical Structures, Vol. 4, No. 2-3 (1996), 129-136)

Comparing my copies of old editions of Bourbaki with latest editions, I
see that he nevertheless sneaks categorical language here and there
without, however, calling it explicitly category theory.

Max

> On 21/05/12 19:49, Staffan Angere 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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Re: Bourbaki & category theory

[Eduardo J. Dubuc]

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.


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Re: Bourbaki & category theory

[rlk]

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


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Re: Bourbaki & category theory

[Eduardo J. Dubuc]

It seems to me that all this discussion concerning whether the notion of
isomorphism comes before of that of morphism or the other way around, or
if Bourbaki had it or not,  is meaningful only on historical basis.

The notion of category as an abstraction of the notion of morphism would
had been forgotten and it would have disappear from mathematical practice.

What happened, as an unexpected consequence of the definition of
category,  was that the notion of universal property found its home, and
may be this is what Bourbaki, which already had the notion of universal
property, missed (I say may be).

This we can see it from the beginning of the first substantial
contributions of categories to mathematics, as for example in the work
of Kan on adjoint functors, or in the fact that a trivial statement as
Yoneda's Lemma is the most important fact in Category Theory.

e.d.

On 23/05/12 08:36, George Janelidze wrote:
> I am ready to take back my criticism and apologize, if the "longer
> sentence" is correct. But is it?
>
> I am certainly not an expert in Bourbaki history, and, as far as I
> remember, they say no word about morphisms in the historical part of
> "Theory of Sets" and give no references on categories. But I think they
> "always" believed that structures determine isomorphisms but not
> morphisms, and I don't think they changed their mind between 1951 and 1957.
>
> When I say "they" I mean "those of them who made main decisions about
> the Bourbaki tractate". Because I hope (!) that not all of them were
> happy that categories are not even defined in the tractate.
>
> In my previous message I wrote "Removing Bourbaki's formalism..." but in
> fact that "formalism" is (not nice but) serious, in the sense that it
> takes us further away from abstract categories.
>
> Anyway, we need to know, if it is still possible, how exactly did
> Bourbaki definition of morphism(s) came up.
>
> George
>

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Re: Bourbaki & category theory

[Colin McLarty]

On Wed, May 23, 2012 at 7:36 AM, George Janelidze <janelg@telkomsa.net> wrote:
> I am ready to take back my criticism and apologize, if the "longer sentence"
> is correct. But is it?

Yes, it is.  I am an expert on Bourbaki history.  See my articles on
Chevalley, Dieudonné, and Weil in the New Dictionary of Scientific
Biography. As Eduardo says, it is a historical question.  Here are the
historical facts

Bourbaki's first publication was

Bourbaki, N. [1939]: Th{\'e}orie des ensembles, Fascicules de
r{\'e}sultats, Paris:
Hermann, Paris.

It is very sketchy on "structures," and uses no notion of mapping
between structures except isomorphisms. Their actual theory of
structures first appeared in

Bourbaki, N. [1957]: Th{\'e}orie des ensembles}, Chapter 4, Paris: Hermann.

That theory was a rear-guard action meant to give an alternative to
category theory.  As i mentioned before, Weil was citing the
categorical idea, and thinking about finding an in-house alternative
to it, already in 1951.  By 1957 Grothendieck, and Cartier, and
Chevalley, probably Dieudonne, and others, all saw that category
theory was more agile than these structure, simpler, and more to the
point, plus it had a natural "higher order" aspect in the theory of
functors which was actually more useful in practice than categories
alone.

Cartier has justly said it would have been a huge job to formulate all
Bourbaki's ideas in terms of categories and functors.  It would have
called for a lot of ideas which were only invented in the coming
years.

It was relatively easy to give Bourbaki's theory of structures --
because it never really worked at all even for Bourbaki's purposes (as
Corry documents in detail).  Naturally it is easier to give an
unusable theory of structures than to work out the ways categories and
functors would actually be used.

best, Colin


>
> I am certainly not an expert in Bourbaki history, and, as far as I remember,
> they say no word about morphisms in the historical part of "Theory of Sets"
> and give no references on categories. But I think they "always" believed
> that structures determine isomorphisms but not morphisms, and I don't think
> they changed their mind between 1951 and 1957.
>
> When I say "they" I mean "those of them who made main decisions about the
> Bourbaki tractate". Because I hope (!) that not all of them were happy that
> categories are not even defined in the tractate.
>
> In my previous message I wrote "Removing Bourbaki's formalism..." but in
> fact that "formalism" is (not nice but) serious, in the sense that it takes
> us further away from abstract categories.
>
> Anyway, we need to know, if it is still possible, how exactly did Bourbaki
> definition of morphism(s) came up.
>
> George
>
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Bourbaki & category theory

[Colin McLarty]

On Wed, May 23, 2012 at 8:03 PM, Eduardo J. Dubuc <edubuc@dm.uba.ar> wrote:
> On 22/05/12 14:49, Eduardo J. Dubuc wrote:
> I received a private mail that can be of interest to many and that I copy
> and paste below:


Following that link, a search leads to

http://mathdoc.emath.fr/archives-bourbaki/PDF/nbt_029.pdf

where we find Le Tribu no. 28 (reporting in a meeting of June-July
1952).  Page 3 includes this report on the state of Theorie des
Ensembles Ch. 4, Structures:

>  Le Congrès à travaillé pour dégager la notion de homomorphisme.

The context is interesting.  The point is, this confirms what was
already known, that Bourbaki worked to find a general notion of
homomorphism after Weil (1951) suggested they should.  And Weil had
named Mac~Lane as suggesting the idea.

best, Colin

.

>
> ===================================================================
> 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.
>

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Bourbaki and category theory again

[George Janelidze]

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




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Bourbaki, Ehresmann & species of structures

[Andree Ehresmann]

Dear all,

Charles Ehresmann was somewhat reticent about Bourbaki's theory of  
structures. In fact, he only participated to the first discussions on  
the "Fascicule de résultats" since he progressively took its distances  
with the group in the late forties. One of the reasons he gave me for  
this withdrawal (among others) was that he thought that a good theory  
of structures should include a theory of "local structures" which has  
motivated a large part of his earlier works.

In several papers in the early fifties, in particular in his 1952 Rio  
de Janeiro course (reprinted in "Charles Ehresmann : Oeuvres completes  
et commentées" Part II-1) he briefly recalls the Bourbaki's  
definition, and then proceeds to develop his own theory of local  
structures. At this time, he did not know the notion of a category,  
the interest of which he discovered later through one of his students.  
It means that categories were not much discussed in France at that  
time...

The first paper where Charles uses categories is the seminal paper  
"Gattungen von lokalen Strukturen", 1957 (reprinted in the same volume  
II-1 of the "Oeuvres"). The aim of the paper is to define a general  
notion of a species of structures and of a species of local structures.
For that, he introduces the action of a category C on a set S, calling  
S a species of structures over C; he proves that it corresponds to a  
functor F from C^op to Set, and defines the associated discrete  
fibration, which he calls a "hypermorphism category".
Then he defines a species of local structures by internalizing this in  
the category of local sets (well before internal categories were  
known…) and gives a "Complete Enlargement Theorem" which translates in  
this categorical setting the construction of the species of local  
structures associated to a pseudogroup of transformations.
Though the definitions are for general categories, in the last parts  
he restricts the categories to groupoids. (This 1957 paper has been at  
the basis of a large part of his/our later works on internal  
categories, sketches, completion theorems,…)

However at that time he said to me that he was not satisfied with the  
notion of structures, and it led him to develop a more categorical  
frame while we were in Buenos-Aires in 1958, introducing the notion of  
   "type functors" (in the paper "Catégories des foncteurs types" 1960,  
reprinted in Volume IV-1 of the "Oeuvres" where is also given the  
first definition of a double category).

His interest of the notion of a species of structures and of  
"covariant maps" between them has motivated the title of his 1965 book  
"Categories et Structures" (Dunod).

Yours
Andree



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Re: Bourbaki and category theory again

[William Messing]

Concerning Borceaux's citation of "Revised and reduced..." in the second 
edition of Bourbaki, Theorie des Ensembles, chapitre IV, I think a more 
accurate translation is "suppressed", rather than "reduced".  What was 
suppressed from the first, 1957, edition of Structures was the 19 page 
appendix, titled Criteres de transportabilite, and  devoted to giving 
useful criteria for verifying that a relation of a theory T, is 
transportable relative to a given typification, as these terms are 
defined in ¶ 1 of this chapter.  Thus, this appendix   also explains the 
concept of "transport of structure", a notion that Bourbaki employed 
frequently and also gives a rigorous definition of "canonical". 

It seems plausible that Chevalley was the author of this appendix, 
although Bourbaki tradition is to not confirm or deny such a statement.

I have never understood why Bourbaki chose to suppress this appendix, as 
I find it very interesting  and very well written.

Bill Messing


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Re: Bourbaki & category theory

[pjf]

A little problem with Fred's note. I never gave a talk at Columbia before
becoming a post-doc in September 1960. (Indeed the embedding theorems hadn't
been proven by either Lubkin or me before September 1959.) But in the
academic year of 1958-59 Lang was giving a course not just at Columbia but
at Princeton (where I was first-year grad student) and it was devoted to
category theory. But from what Sammy Eilenberg later told me, Lang was
certainly broadcasting the stuff I was doing. (My undergrad honors thesis
at Brown had contained the special case of the special and general
adjoint functor theorems, to wit, the case where one of the two
functors is an
inclusion functor. I learned only later about adjoints -- the proofs for
my reflective and coreflective subcategories worked without change.) Sammy
often complained to me in the following years how Lang told everybody that
I was the world's greatest category expert.



Quoting "Fred E.J. Linton" <fejlinton@usa.net>:

> ...in the first-year graduate algebra course [Serge Lang] gave at Columbia
> during the 1958-1959 academic year...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":-) ).


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Re: Bourbaki & category theory

[Fred E.J. Linton]

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



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