Re: Bourbaki & category theory

["George Janelidze" <janelg@telkomsa.net>]

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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