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:
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>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 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/