Re: Bourbaki & category theory

[Colin McLarty <colin.mclarty@case.edu>]

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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