Bourbaki, Ehresmann & species of structures

[Andree Ehresmann <andree.ehresmann@u-picardie.fr>]

Dear all,

Charles Ehresmann was somewhat reticent about Bourbaki's theory of  
structures. In fact, he only participated to the first discussions on  
the "Fascicule de résultats" since he progressively took its distances  
with the group in the late forties. One of the reasons he gave me for  
this withdrawal (among others) was that he thought that a good theory  
of structures should include a theory of "local structures" which has  
motivated a large part of his earlier works.

In several papers in the early fifties, in particular in his 1952 Rio  
de Janeiro course (reprinted in "Charles Ehresmann : Oeuvres completes  
et commentées" Part II-1) he briefly recalls the Bourbaki's  
definition, and then proceeds to develop his own theory of local  
structures. At this time, he did not know the notion of a category,  
the interest of which he discovered later through one of his students.  
It means that categories were not much discussed in France at that  
time...

The first paper where Charles uses categories is the seminal paper  
"Gattungen von lokalen Strukturen", 1957 (reprinted in the same volume  
II-1 of the "Oeuvres"). The aim of the paper is to define a general  
notion of a species of structures and of a species of local structures.
For that, he introduces the action of a category C on a set S, calling  
S a species of structures over C; he proves that it corresponds to a  
functor F from C^op to Set, and defines the associated discrete  
fibration, which he calls a "hypermorphism category".
Then he defines a species of local structures by internalizing this in  
the category of local sets (well before internal categories were  
known…) and gives a "Complete Enlargement Theorem" which translates in  
this categorical setting the construction of the species of local  
structures associated to a pseudogroup of transformations.
Though the definitions are for general categories, in the last parts  
he restricts the categories to groupoids. (This 1957 paper has been at  
the basis of a large part of his/our later works on internal  
categories, sketches, completion theorems,…)

However at that time he said to me that he was not satisfied with the  
notion of structures, and it led him to develop a more categorical  
frame while we were in Buenos-Aires in 1958, introducing the notion of  
   "type functors" (in the paper "Catégories des foncteurs types" 1960,  
reprinted in Volume IV-1 of the "Oeuvres" where is also given the  
first definition of a double category).

His interest of the notion of a species of structures and of  
"covariant maps" between them has motivated the title of his 1965 book  
"Categories et Structures" (Dunod).

Yours
Andree



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