quotients of groupoids

[<K.Mackenzie@sheffield.ac.uk>]

This is a comment on one aspect of Jean Pradines' very interesting
posting of June 8. I quote a large part of it for reference:

>>>>>>>>>>>>>>>>>>>>>>>>>>>>

EXTRACT FROM JEAN'S POSTING:

Now what I really had in mind was not the purely algebraic (rather obvious)
setting, but the study of quotients for the differentiable groupoids
(nowadays called smooth or Lie) introduced by Ehresmann in 1959, which are
groupoids in the category of manifolds.
The statement I had obtained (which again certainly cannot by any means be
deduced from the general results of Ehresmann concerning topological or
structured groupoids) was published in 1966 in my Note (prealably submitted to Ehresmann and transmitted by him) :

-Théorie de Lie pour les groupoïdes différentiables. Relations entre
propriétés locales et globales, CRAS (Paris), série A, t.263, p.907-910, 19
décembre 1966-

in Théorème 5 (in this statement the Bourbaki term "subimmersion" is
improperly used and has to be understood in the more restricted acception
"submersion onto a submanifold" ; note also that the implication from 2° to 1° is valid only
when the fibres of the domain map of H are connected). (Note that in this statement the algebraic theory of the two-sided quotient of a groupoid by an invariant subgroupoid is implicitely considered as "well known" without reference. This was in fact a diplomatic consequence of the above-mentionned conversation ! At least at that period it seems absolutely certain that no reference did exist, and probably very few people, if any, had had the opportunity of thinking to that sort of questions, since the main stream of categoricists were despising groupoids as beig trivial, since equvalent to groups).
The unpublished proof of this statement relied on a careful and rather
delicate study of the foliation defined by the two-sided cosets.
The very elegant proof for the case of Lie groups given in Serre, Lie
Algebras and Lie Groups (lecture in Harvard, 1964, p.LG 4.10-11), relying on the so-called Godement's
theorem, works only for one-sided cosets and yields only the too special case
above-mentioned or more generally the statement of Proposition 2 in the
previous Note (which extends to groupoids the classical theory of homogeneous spaces for possibly non invariant subgroups).

It is clear that this last proof may be immediately written in a purely
diagrammatic way and remains valid in much more general contexts when an
abstract Godement's theorem is available. It turns out that this is the case
for most of (perhaps almost all)  the categories considered by "working
mathematicians", notably the abelian categories as well as toposes, the
category of topological spaces (with a huge lot of variants), the
category of Banach spaces, and many useful categories that are far from being complete.The precise definitions for what is meant by an abstract Godement's theorem were given in my paper :

-Building Categories in which a Godement's theorem is available-
published in the acts of the Second Colloque sur l'Algèbre des Catégories,
Amiens 1975, Cahiers de Topologie....(CTGDC).

In this last paper I introduced the term "dyptique de Godement" for a
category in which one is given two subcategories of "good mono's" and "good
epi's" (playing the roles of embeddings and surmersions in Dif) such that a formal Godement's theorem is valid.

These considerations explain why I was strongly motivated for adapting
Serre's diagrammatic proof to the case of two-sided cosets (since such a proof would immediately extend the theory of quotients for groupoids in various non complete categories used by "working mathematicians", yielding a lot of theorems completely out of the range of Ehresmann's general theory of structured groupoids). However this was
achieved only in 1986 in my Note :

-Quotients de groupoïdes différentiables, CRAS (Paris), t.303, Série I, 1986, p.817-820.-

The proof requires the use of certain "good cartesian squares" or "good pull back
squares", which, though they are not the most general pull back's existing
in the category Dif of manifolds, cannot be obtained by the (too
restrictive) classical condition of transversality. Though written in the
framework of the category Dif (in order not to frighten geometers, but with
the risk of frightening categoricists), this paper is clearly thought in
order to be easily generalizable in any category where a suitable set of
distinguished pull back's is available , assuming only some mild stability properties.

In this paper I introduced the seemingly natural term of "extensors" for
naming those functors between (structured) groupoids which arise from the
canonical projection of a groupoid onto its quotient by a normal
subgroupoid. Equivalent characterizations are given.(I am ignoring if another term is being used in the literature).This notion is resumed and used in my paper :

-Morphisms between spaces of leaves viewed as fractions-
CTGDC, vol.XXX-3 (1989),p. 229-246

which again is written in the smooth context, but using purely diagrammatic descriptions (notably for Morita equivalences and generalized morphisms) allowing immediate extensions for various categories.
As a prolongation of this last paper, I intend in future papers to give a
description of the category of fractions obtained by inverting those
extensors with connected fibres. This gives a weakened form of Morita
equivalence which seems basic for understanding the transverse structure of
foliations with singularities.

<<<<<<<<<<<<<<<<<<<<

In response to Jean's paper

[QGD] Quotients de groupoïdes différentiables, CRAS (Paris), t.303, 
Série I, 1986, p.817-820. 

Philip Higgins and I wrote two papers ([HM90a] and [HM90b] below) 
dealing with general quotients of Lie (=differentiable) groupoids 
and Lie algebroids. Our starting point was the idea that:

``a morphism in a given category is entitled to be called a quotient 
map if and only if it is entirely determined by data on the domain''

(Perhaps this will seem naive to true categorists, but I write as an 
end user of category theory, not a developer.) 

I would summarize [QGD] as proving that the `regular extensors' are
quotient maps in the category of Lie groupoids. [HM90b] then showed,
by extending the notion of kernel, that in fact all extensors are
quotient maps in the category of Lie groupoids.

Philip and I used the term `fibration' for what Jean calls an
`extensor'; these maps satisfy a natural smooth version of the 
notion of `fibration of groupoids' introduced by Ronnie Brown 
in 1970 (building on work of Frolich, I think). (Throughout this 
post, I assume base maps to be surjective submersions.)

There is unlikely to be a more general class of quotient maps for
Lie groupoids: the fibration condition on a groupoid morphism 
$F : G \to H$ is exactly what is needed to ensure that any product 
in the codomain is determined by a product in the domain: given 
elements $h, h'$ of $H$ which are composable, one wants to be able 
to write $h = F(g)$ and $h' = F(g')$ in such as way that $gg'$ will 
exist and determine $hh'$; the fibration condition is the weakest 
simple condition which ensures this. 

Fibrations of Lie groupoids are not determined by their kernel in 
the usual sense (= union of the kernels of the maps of vertex 
groups); one also requires the kernel pair of the base map, and the 
action of this (considered as a Lie groupoid) on the manifold of 
one--sided cosets of the domain. This data, which Philip and I 
called a `kernel system' is equivalent to a suitably well--behaved
congruence on the domain groupoid. Though more complicated than the
usual notion of kernel, the notion of normal subgroupoid system 
gives an exact extension of the `First Isomorphism Theorem'. 

We also showed that regular fibrations (= Jean's regular extensors)
are precisely those in which the additional data can be deduced from
the standard kernel and the base map; in the regular case the two 
step quotient can be reduced to a single quotient consisting of 
double cosets (as Jean remarks in his post). The congruences 
corresponding to regular fibrations are those which, regarded as 
double groupoids, satisfy a double source condition. 

A good example of a fibration which is not regular is the division 
map $\delta: (g,h)\mapsto gh^{-1}$ in a group $G$, considered as a 
groupoid morphism from the pair groupoid $G\times G$ to the group $G$.

All of this (for Lie algebroids, as well as for Lie groupoids and 
for vector bundles) is in the two papers referenced below. There is 
also a full account coming in my book `General Theory of Lie 
groupoids and Lie algebroids' (CUP) which should be appearing 
in the next few months. 


@ARTICLE{HM90a,
   author = {P.~J. Higgins and K.~C.~H. Mackenzie},
   title = {Algebraic constructions in the category of {L}ie 
		algebroids},
   journal = {J.~Algebra},
   year = 1990,
   volume = 129,
   pages = "194-230",
}

@ARTICLE{HM90b,
   author = {P.~J. Higgins and K.~C.~H. Mackenzie},
   title = {Fibrations and quotients of differentiable groupoids},
   journal = {J.~London Math. Soc.~{\rm (2)}}, 
   year = 1990,
   volume = 42,
   pages = "101-110",
}

Kirill Mackenzie
http://www.shef.ac.uk/~pm1kchm/