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.
>>>>>>>>>>>>>>>>>>>>>> =2E........................................................................= =2E.. =2E........... <<<<<<<<<<<<<<<<<<
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/