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