Some history on quotient categories

Some history on quotient categories

[Andree Ehresmann]

Here are some historical notes, in complement to Pradines' comments on 
multiplicative graphs and on quotient categories.

Charles Ehresmann introduced multiplicative graphs as an algebraic support 
to the notion of a "category kernel", which corresponds to the structure 
induced on an open subset of a topological category (in the sense of 
Charles, i.e. a category internal to Top). Such category kernels generalize 
group kernels (that Charles used in the second part of his thesis to define 
locally homogeneous spaces); we had considered  them in the early sixties 
both in his papers on differential geometry and in mine on optimization 
problems. Then multiplicative graphs appeared as a natural tool in his 
study on quotient categories.

Charles came to quotient categories by his works on foliations and on 
prolongations of manifolds. In his long 1962 paper on foliations (1), he 
constructs several kinds of holonomy groupoids by a quotient process. In 
his paper on topological categories (2), he construct categories of jets 
(local jets and infinitesimal jets) as quotient categories of the category 
of local sections of a topological category. In fact he had exposed these 
constructions much earlier in his lectures and briefly alluded to in his 
papers on differential geometry in the late fifties.

So he was naturally led to a more formal study of quotients, which he began 
in his paper "Structures quotient" (3), where in particular he introduces 
the "strict quotient"; this paper (abstracted in (4)) is more easily read 
than later papers. In a comment I have added to this paper in the "Oeuvres" 
(comment 170, p. 375) I mentioned other authors who have studied quotient 
categories about the same time, in particular Dedeckerand Mersch, Higgins, 
Hoenke, Pümplun.

Later on, Charles tried to develop a non-abelian cohomology for which he 
wanted to define a natural notion of "short sequence" in Cat. For this he 
needed to define quotients of a category or of a groupoid by a 
sub-category. It is done in the paper (5). The main results of this paper 
are taken back in his book "Categories et Structures" (Dunod 1965), with 
some illustrative diagrams (it is to this book that Pradines refers).
In several papers, he generalized the theory of quotient categories in the 
frame of internal categories (cf. "Oeuvres", Parts III and IV), .

1. "Structures feuilletees", Proc. 5th Canadian  Math. Congress; reprinted 
in "Charles Ehresmann, Oeuvres completes et commentees" Part II-2, 563-626.
2. "Categories topologiques, III", Indig. Math. 28, 1966; reprinted in the 
"Oeuvres" Part II-2, 655-669.
3. "Structures quotient",.Comm. Math. Helv. 38, 1963; reprinted in 
"Oeuvres' III-1, 143-293.
4.  "Structures quotient et cegories quotient", CRAS, 256, 1963, 5031-34, 
reprinted in "Oeuvres" III-1, 9-11.
5. "Cohomologie dans une categorie dominee", Proc. Coll. Topologie, CBRM 
1964; reprinted in "Oeuvres" III-2, 531-590.