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