Re: reference : normal categorical subgroup ?
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
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/
This is a purely terminological remark concerning the comments sent by Kirill Mackenzie about the response I made to Marco Mackaay's message ("reference normal categorical subgroup ?", June 5). From these comments, fully quoted below (though omitting the partial quotations from my own message of June 8), I extract the following two fragments : <<Philip and I used the term `fibration' for what Jean calls an`extensor'>> <<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>> Indeed Kirill makes a double confusion (probably caused by the ellipticity of the redaction of my Note referenced below as [QGD]) about the meanings I attribute to the terms "extensor" and "regular". As to the latter term, it has for me a purely smoothness meaning (referring to the fact that the equivalences on the manifolds have to be regular ones, and also that the anchor map of the kernel has to be a regular morphism, by which I mean composed of a surmersion and an embedding) and this term has to be dropped when dealing with the purely algebraic aspect of the question. In my paper referenced [MVF] I use instead (in view of more generality) the term "s-extensor", where the prefix "s-" refers to the more general context of "diptych" data (in which goodepics/good monos generalize and replace surmersions/embeddings ; see my previous message), and, in the algebraic context, has to be read as just meaning "surjective" ( but is then considered as implied by the very term "extensor"). On the opposite the term "extensor" has for me a purely algebraic meaning (more restrictive than what is called "fibration" in Kirill's message) and is equivalent to the notion of what is called "regular fibration" in Kirill's message (where here "regular" is given an algebraic meaning which I never used for myself !), and refers to the very simple case, alluded to in my own previous message, where one can mimic exactly the theory of group extensions or surjective group homomorphisms (with the only caution of using two-sided cosets). The suffix "or" is to remind that this a funct-or instead of a funct-ion. (In the absence at the present day of any response from Marco, I guess, without being sure, that he was probably interested mainly in this case, or perhaps in the still more special case when the kernel reduces to a sum of groups ). It should be noticed that the (rather obvious) examples given at the end of my Note emphasize the independance of the algebraic conditions (extensor) and the smoothness ones (regular), so that the ambiguity, if it existed, should be cleared up for the careful reader (in the last of these examples the underlying algebraic functor is just the identity, but not the identity for the underlying smooth structures). The so-called "fibrations" were out of the scope of [QFD], which was centred on the smoothness questions and not on the (obvious in that special case) algebraic aspect, implicitely (and perhaps imprudently !) considered there as "well known". However they are considered, and play a basic role, in [MVF] (p. 238 § 7) under the name of "(surjective) exactors" (explained below), but the general problem of quotients and generalized kernels in the sense described and referenced by Kirill (which is of course much more delicate than the algebraically obvious case of extensors) is completely out of the scope of this paper. The remark (ii) of this page 238 emphasizes the fact that "extensor" implies "surjective exactor". Now it should be clear that when dealing with topological or smooth (i.e. Lie) groupoids, terms such as "(regular) fibrations" have to be definitely rejected, though previously used by various authors in the purely algebraic (or categorical) context, as giving rise to unsolvable ambiguities. (Note that there is a similar problem when using the widespread terminology "discrete" and "coarse" for groupoids ; though the ambiguity is generally much less disastrous in that case, I think it much better to use respectively the terms "null" and "banal"). The reason is of course that these terms have very ancient (various !) meanings in Topology and Differential Geometry, which are not at all implied by (nor imply) the algebraic condition, nor by weaker topological or smoothness conditions such as (surjective) open maps or surmersions. I remind that unhappily the term "foncteur fibrant" was introduced very early by Grothendieck and his school, and is, I think, still used by most category theorists, concurrently with the term "fibration", which appeared, I think, a little later. Ronnie Brown used also sometimes the more suggestive and non ambiguous term "star-surjective", which might become "star-surmersive" in the smooth case (and perhaps something like "star-epic" in my more genaral dyptich framework, though I don't intend to use it). The terminology proposed in [MVF] (to which I refer for more precision and details omitted here) comes from a general analysis of the properties of a functor f between two groupoids, going from H to G (here we shall always assume below, to simplify, that f induces a surjective map for the bases, thus omitting as a consequence the word "surjective" in many occurences in what follows ; see [MVF] for more general and precise definitions and statements). Forgetting for a while (to make the things simpler) the smoothness (or diptych) framework to consider solely the algebraic properties, I believe that the most important of these are reflected by the two commutative squares a(f) and t(f) built, from f, respectively with the domain maps and the anchor maps (of H and G), and more precisely by the properties of injectivity/surjectivity/bijectivity of the two canonical arrows (denoted below by u and w) going from H to the pull back's generated by these two squares. (My general guess and philosophy is then that the suitable corresponding notions in the smooth or more generally "diptych" case -see my previous message- are gotten by just replacing "injective/surjective/bijective" by "good mono/good epi/iso", and that as soon as one is able to describe the set theoretic algebraic definitions, constructions and proofs by means of diagrams, everything extends "almost automatically" to the structured case, using the Godement diptych axioms). In that context the extensors are just defined very quickly by the surjectivity of w and the "exactors" (here always surjective)by the surjectivity of u ("star-surjectivity" in the sense of Ronnie). (Note that the bijectivity of w characterizes the surjective equivalences, which are special instances of extensors). Now it turns out readily that the bijectivity of u characterize those functors which describe actions of the groupoid G on the base of H (H is then called the action groupoid in the literature, but I emphasize the fact that the action of G is not fully described by H alone, but by the functor f). For that reason I believe quite natural (though I don't seem to be followed) to call "(surjective) actors" the functors of this type (note that the classical terminology in categorical works is "discrete fibrations" (!), "foncteurs d'hypermorphismes" (!!) in Ehresmann's book, and sometimes "star-bijective" for Ronnie Brown). This explains (but perhaps does not justify) the above-mentioned term "exactor", with the suffix "or" as supra, and the prefix "ex" supposed to remind the surjectivity property of u (and not some terrorist or prejudicial activity) while evocating also some generalized kind of ex-tension. The (surjective) actors and extensors appear as two opposite ways of degenerating for the (surjective) exactors, while the theory of Kirill and Philip explains how these two special cases are mixed up in the general (more sophisticated) case. There is also a very interesting special case of exactors described in [MVF] under the name of "subactors" (Prop.-Def. 7.5), which are the faithful ones. They make up a subcategory whose arrows admit a unique factorization through a surjective equivalence and an actor. (As a general remark all the purely algebraic underlying content of [MVF] may be considered as more or less easy or even trivial and or more (or less ?) well known, but again the interesting point is that the rather easy set-theoretical proofs may be (with some care) written diagrammatically in order to be transferred to the smooth case via the diptych method). In a secund part, I'll add some other terminological remarks about the terminology of "quotient groupoids" used by Kirill. Jean PRADINES References (J. Pradines) [QGD] Quotients de groupoïdes différentiables, CRAS (Paris), t.303, Série I, 1986, p.817-820. [MVF] Morphisms between spaces of leaves viewed as fractions, CTGDC (Cahiers de Topologie.....), vol.XXX-3 (1989),p. 229-246 ----- Original Message ----- From: <K.Mackenzie@sheffield.ac.uk> To: Categories List <categories@mta.ca> Sent: Tuesday, July 01, 2003 11:33 AM Subject: categories: quotients of groupoids 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:
>>>>>>>>>>>>>>>>>>>>>> .................................................................
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 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/
This is is the announced secund part added to my response to Kirill's comments concerning terminology. The first part deals with the relation between my "regular extensors" and Kirill's "regular fibrations", and points a misunderstanding. This secund part cannot be read independently of the first one and will concern Kirill's use of the term "quotient of groupoids".
From his message quoted below, I now extract the following :
<<``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.) >> <<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 $h =3D F(g)$ and $h' =3D F(g')$ in such as way that $gg'$ will exist and determine $hh'$; the fibration condition is the weakest simple condition which ensures this. >> First I observe that the expression "entirely determined by data on the domain'' can hardly be given a precise mathematical or logical status, sinc= e it seems difficult to justify in a formalisable way why the (tautological) data of the given functor f (from H to G) itself could not be considered as data "on H", whil= e the extra data of an equivalence relation, a subgroupoid, or a "kernel system", would be such. On the other hand a morphism such as,for instance, the diagonal map H --> H x H is certainly entirely determined by the single data H without any extra data != ! Precise definitions of the various possible notions of quotient categories and a thorough discussion can be found in Ehresmann's book, at least in theory. The case of groupoids is treated as a special case, but, as pointed in my first message to Marco, Ehresmann always missed the very simple and important (though or because) special case of what I call extensors ( =3D regular fibrations). This study is made for categories and also for a weaker notion called "multiplicative graphs" (in which the associativity is dropped and the composite of two adjacent arrows may be undefined). The stubborn Ehresmann's reader might perhaps guess that this notion was probably introduced precisely in order t= o handle the composition laws one gets when studying equivalence relations on a category which are compatible in a natural sense with the algebraic data. When such a quotient law defines a true category law (which is not automatic), this one is called a "strict quotient". There exist quotients which are not strict, but satisfy the natural universal property of quotients.The condition of what I call "exactors" (=3D fibrations) is given as sufficient for a quotient being strict. However it is very difficult to extract these (interesting and important) facts from the book because of the very clumsy notations, terminology and redaction, and the (intentional !) absence of examples. So, instead of asking the reader to try to read Ehresmann's book, it will be enough for a clear understanding to give some very simple examples of what may happen, being content with the case of groupoids, in order to illustrate Ehresmann's definitions and their motivations (I confess that I have no idea of the examples Ehresmann had in mind, and I don't claim I understood everything ! ). First we give an alternative definition for strict quotients as resulting from criteria proved in Ehresmann's book : a quotient is strict when the map Vf : VH --> VG (where the functorial symbol V is used here, unusually, for denoting composable pairs or pairs with the same domai= n or the same codomain) is surjective (this allows an immediate transfer to the smooth case via the diptych policy). In the following examples, we'll denote by I (resp. D (written for Delta, pictured by a triangle)) the banal (=3D coarse) groupoid with just two (res= p. three) objects. FIRST EXAMPLE : Let H be the groupoid III (sum of three copies of I), G =3D D, and f be the map sending the three copies of I respectively onto the three edges of the triangleD. (Note that all these groupoids may be considered as topological or even smooth withe the discrete topology, and hence f as continuous or smooth).Then : -1) f defines a surjective functor ; let R be the equivalence relation on H thus defined. -2) f is faithful, but not full, and not an exactor. -3) R is "bicompatible" in Ehresmann's sense, wich means compatible with th= e composition law of H as well as with the source and range (otherwise domain and codomain) maps. -4) The quotient law thus defined on G is only a part of the groupoid law (it defines a "multiplicative graph" in the sense of Ehresmann). So H is not a "strict quotient" in Ehresmann's sense. -5) However this groupoid law is "entirely determined" by the quotient law (hence by H and R) in that sense that it is the unique groupoid law extending that quotient law and defining a quotient in the universal sense (more precise statements can be found in Ehresmann's book). So f should be called a quotient map by Kirill according to the first quotation (one does not see why the data of the equivalence relation R,whic= h determines eveything, could not be considered as data "on H"). But this contradicts the secund one. -6) (f,G) is a quotient groupoid of H in the very general and widely admitted (not only by Ehresmann, but by Bourbaki and nearly everybody) sense of quotient structures, which means here that it satisfies the following universal property of quotients : given any groupoid Z and any functor h from H to Z admitting a set theoretical factorization h =3D gf, then g defines a functor from G to Z. [Exercise : show that f is composed of an (injective) groupoid equivalence, and a surjective actor. Hint : embed suitably H =3D III into K =3D DDD (sum= of three copies of D), and note that K may be viewed as an action groupoid.] SECUND EXAMPLE : We now define H by suppressing a copy of I in the previous example and restricting f. Nothing is changed in the conclusions save the surjectivity of f, so that now G is no more set-theoretic quotient, but is still a quotient in the categorical sense. THIRD EXAMPLE : We start again with III, but define H by now adding a copy of D and extending f obviously (i.e. by the identity). Then what is changed in the conclusions is that now the groupoid law of G may be fully defined as the quotient law (since composable arrows of G are now always images of some composable arrows of H), so that (f,G) is a "strict quotient" of H in the sense of Ehresmann, though f is not an exactor. Once again it should be a quotient in the sense of Kirill according to the first (informal) "definition", though it still contradicts the secund one. So we see that we have a chain of strict implications : quotient (in the universal sense) <=3D=3D surjective quotient <=3D=3D stric= t quotient <=3D=3D surjective exactors (alias fibrations) <=3D=3D extensors (= alias regular fibrations). I think there is absolutely no reason to keep the word "quotient map", for the fourth term of this chain, which is just a criterion for quotients amon= g others. Moreover among the exactors (star surjectivity condition alone), surjective exactors (where the surjectivity is a consequence of the surjectivity for the bases) are just a special case : the importance of exactors comes mainly, I think, from the fact that any functor f is isomorphic with an exactor (which is surjective iff f is essentially surmersive) (see Prop. 8.1, 8.2 and 10.4 of [MVF]) and all of this extends to the smooth , via the general "diptych" policy. JEAN PRADINES ----- Original Message ----- From: <K.Mackenzie@sheffield.ac.uk> To: Categories List <categories@mta.ca> Sent: Tuesday, July 01, 2003 11:33 AM Subject: categories: quotients of groupoids This is a comment on one aspect of Jean Pradines' very interesting posting of June 8.
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In response to Jean's paper [QGD] Quotients de groupo=EFdes diff=E9rentiables, CRAS (Paris), t.303, S=E9rie I, 1986, p.817-820. Philip Higgins and I wrote two papers ([HM90a] and [HM90b] below) dealing with general quotients of Lie (=3Ddifferentiable) 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 $h =3D F(g)$ and $h' =3D F(g')$ 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 (=3D 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 (=3D 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 =3D {P.~J. Higgins and K.~C.~H. Mackenzie}, title =3D {Algebraic constructions in the category of {L}ie algebroids}, journal =3D {J.~Algebra}, year =3D 1990, volume =3D 129, pages =3D "194-230", } @ARTICLE{HM90b, author =3D {P.~J. Higgins and K.~C.~H. Mackenzie}, title =3D {Fibrations and quotients of differentiable groupoids}, journal =3D {J.~London Math. Soc.~{\rm (2)}}, year =3D 1990, volume =3D 42, pages =3D "101-110", } Kirill Mackenzie http://www.shef.ac.uk/~pm1kchm/
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K.Mackenzie＠sheffield.ac.uk