From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2377 Path: news.gmane.org!not-for-mail From: "jpradines" Newsgroups: gmane.science.mathematics.categories Subject: Re: quotients of groupoids (K. Mackenzie's comments on J. Pradines'answer to Marco Mackaay) Date: Sat, 5 Jul 2003 00:55:02 +0200 Message-ID: <002301c342df$11655660$edd8f8c1@wanadoo.fr> References: Reply-To: "jpradines" NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241018614 3832 80.91.229.2 (29 Apr 2009 15:23:34 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:23:34 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Sun Jul 6 13:22:47 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 06 Jul 2003 13:22:47 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19ZCDM-0007Ye-00 for categories-list@mta.ca; Sun, 06 Jul 2003 13:18:12 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 5 Original-Lines: 264 Xref: news.gmane.org gmane.science.mathematics.categories:2377 Archived-At: 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 : <> <> 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 =A7 = 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". =20 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. =20 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).=20 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).=20 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.=20 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 groupo=EFdes diff=E9rentiables, CRAS (Paris), t.303, S=E9rie 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: To: Categories List 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 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/