reference: normal categorical subgroup?

reference: normal categorical subgroup?

[Marco Mackaay]

To all category theorists,

I'm looking for a reference to the definition of a normal categorical
subgroup, i.e. the right kind of subgroupoid of a categorical group for
taking the quotient. I know the definition, but I have no reference. Does
anyone know a published origin of the definition?

Best wishes,

Marco Mackaay

Re: reference: normal categorical subgroup?

[Ronnie Brown]

A definition of normal subcrossed module was given by Kathy Norrie in
her thesis, (see the references to her work in two papers in the April,
2003, issue of Applied Categorical Structures). Of course crossed modules
are equivalent to cat^1-groups, (G,s,t), so it is a nice exercise to do
the translation.  The expected answer is (H,s',t') such that H is normal
in G and invariant under s,t.

However categorical groups are different to cat^1-groups, so it is a nice
exercise to solve this problem. It would also be good to characterise
groupoid objects internal to categorical groups, since these
should generalise the congruences for those objects, and hopefully lead to
a definition of free groupoid in categorical groups, and so to notions
of free resolutions.

For cat^1-groups (and crossed modules) there is a nice notion of induced
object (see papers by Brown and Wensley in TAC, for example) which is
required to compute the 2-type of a mapping cone induced on classifying
spaces by a morphism of groups.

Best regards

Ronnie


On Thu, 5 Jun 2003, Marco Mackaay wrote:

> To all category theorists,
>
> I'm looking for a reference to the definition of a normal categorical
> subgroup, i.e. the right kind of subgroupoid of a categorical group for
> taking the quotient. I know the definition, but I have no reference. Does
> anyone know a published origin of the definition?
>
> Best wishes,
>
> Marco Mackaay
>
>
>
>
>
>
>
>

Prof R. Brown,
School of Informatics, Mathematics Division,
University of Wales, Bangor
Dean St., Bangor,
Gwynedd LL57 1UT, United Kingdom
Tel. direct:+44 1248 382474|office:     382681
fax: +44 1248 361429
World Wide Web:
home page: http://www.bangor.ac.uk/~mas010/
(Links to survey articles:
Higher dimensional group theory
Groupoids and crossed objects in algebraic topology)

 Centre for the Popularisation of Mathematics
 Raising Public Awareness of Mathematics CDRom
 Symbolic Sculpture and Mathematics:
 http://www.cpm.informatics.bangor.ac.uk/centre/index.html

Re: reference : normal categorical subgroup ?

[jpradines]

Though I have no reference for just the definition, I can make some
historical remarks which perhaps explain the absence of reference, and which
may be of interest for some people, and also give references for some (much
less obvious) developments of the subject concerning groupoids in the
category of manifolds and much more general ones.

About 1964 or 5, I had the opportunity of mentioning to Charles Ehresmann
that I had noticed the fact that the theory of factoring a group by a normal
(or invariant or distinguished) subgroup extends almost obviously to groupoids.
I was very surprised with his answer : "The question of quotients is a very
difficult one which I have solved recently and a part of the theory will be
proposed as the subject of examination for my students of DEA. I really
don't know how you could manage". It is likely that this was a polite way of
suggesting that I was certainly wrong in his opinion, but he didn't want to
listen more explanation.
Sometimes later, reading chapter III of his book on categories (published in
1965), I realized that he was certainly alluding to his very general theory of
factoring a category by an equivalence relation or by a subcategory, while satisfying a universal property. Though
this theory looks rather exhaustive and contains some rather deep and
sophisticated statements, it seems in my opinion strictly impossible to
deduce from any of these statements the very simple case of groupoids and
normal subgroupoids nor even the very definition of a normal subgroupoid.
I just recall here briefly what has certainly been discovered (at least partially) by any people
having had the opportunity of meeting the question and thinking ten minutes
to it, to know that the case of groups can be exactly mimicked for groupoids with
just two precautions :
-first, of course, expressions such as xHy have to be understood as denoting
all the composites of the form xhy which are defined (where h runs in H) ;
as a consequence the condition of normality for H in G (in which y is the inverse of x) bears only on the
isotropy groups of H;
-secund and more significantly it is no more true in general that the right and left
cosets xH and Hx coincide ; however one just has to define the elements of the factor
groupoid as consisting of two-sided cosets HxH ; with this slight modification of the
theory, everything becomes an obvious exercise.
One has also to notice that there is a special case where no modification is
required, to know the case when H is actually a subgroup, i.e. just reduced
to its isotropy groups. In that case the base of the factor groupoid is not
changed. This case is the most obvious and also probably the best known (and
possibly for most people the only one known), but  in my opinion certainly
not the most interesting, since it does not include the quotient of a set by an equivalence relation, whose graph is viewed as a subgroupoid of a coarse groupoid.

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.

                             Jean PRADINES

---- Original Message -----
From: Marco Mackaay <pmzmm@mat.uc.pt>
To: categories <categories@mta.ca>
Sent: Thursday, June 05, 2003 4:49 PM
Subject: categories: reference: normal categorical subgroup?


> To all category theorists,
>
, p.> I'm looking for a reference to the definition of a normal categorical

reference: normal categorical subgroup?

[Marco Mackaay]

To all category theorists,

I'm looking for a reference to the definition of a normal categorical 
subgroup, i.e. the right kind of subgroupoid of a categorical group for
taking the quotient. I know the definition, but I have no reference. Does
anyone know a published origin of the definition?

Best wishes,

Marco Mackaay