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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