Re: Bourbaki and Categories

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

Dear Colleagues,

I think the first things to say about "Bourbaki and Categories" are:

(a) It is very obvious that the invention of category theory was by far the
greatest discovery of 20th century mathematics.

(b) Bourbaki Tractate is another great event, of a very different kind of
course, which will be a treasure for the Historians of next centuries. It
shows how the members of a very leading group of a leading mathematical
country were thinking in the middle of the same century (well, up to their
internal disagreements; after all, Eilenberg and Grothendieck were also
there at some point...).

(c) Accordingly, Bourbaki Tractate is the best evidence showing how hard it
was to understand (even and especially for such brilliant mathematicians!)
that there is something even better that Cantor paradise.

(d) Defining structures, Bourbaki makes very clear that morphisms are
important (and some form of universal properties are important). But
morphisms are NOT defined in general: it is simply a class of maps between
structures of a given type closed under composition and having isomorphisms
(which ARE defined) as its invertible members. And... every interested
student will ask: if so, why not defining a category?

Let me also add what is less important but still comes to my mind:

(e) Bourbaki approach to structures has a hidden very primitive form of what
was later discovered by topos theorists: in order to define a structure they
need a 'scales of sets', which is build using finite products and power sets
(no unions and no colimits of any kind!).

(f) According to Walter Tholen's message, Karl Heinrich Hofmann says: "...it
is truly surprising that the theory of categories (S. Eilenberg and S. Mac
Lane, 1946) was almost demonstrably
ignored as the mother of all structure theories. This was hardly sustainable
in commutative algebra anymore...". Very true (except 1946), but it is
much-much-much worse in homological algebra, where the absence of categories
and functors (having a section called "Functoriality" though) in Bourbaki's
presentation is most amazing.

(g) A few days ago Tom Leinster has explained to us that "disinformation is
*deliberate* false information, false information *intended* to mislead".
Fine, but sometimes false information is created by ignorance so badly, that
it sounds right to call it disinformation (Don't you agree, Tom?). And...
look at http://en.wikipedia.org/wiki/Bourbaki : There is a section called
"Criticism of the Bourbaki perspective", which, among other things, says:
"The following is a list of some of the criticisms commonly made of the
Bourbaki approach:^[13]..." (where [13] is a book of Pierre Cartier; I have
not seen that book, and so I am not making any conclusions about it). The
list has seven items with no category theory in it!

George Janelidze