ARRINDELLZ@aol.com writes:
Hi,Prof.(I having trouble posting msg on the net) so maybe you can help, a Categorist whan-na-be.
I read somewhere on the web that "Monodromy is a general concept in category theory involving the globalization of local morphism." I cannot find any work in Categorical Monodromy.
Do you know of any Categorical Monodromy paper I can read?
Thanks
That quote looks straight form Mathworld :-). It seems a bit misleading. Monodromy comes to us from classical complex analysis (where it is about uniqueness of analytic continuation) and has been greatly generalized to
point of becoming "a general concept ... involving the globalization of local morphism." But this is not about category theory per se, but rather a variety of applications. In any case here are a few pointers to information on
reply to r.brown@bangor.ac.uk Thanks Robert for mentioning my work. It may be useful to state the key ideas, which I learned from Pradines in the early 1980s, and which go back to the first of his 4 CRAS notes in the 1960s, which state results. The idea of monodromy as extension of local morphisms goes back to the book on Lie groups by Chevalley, which deals with the case of an equivalence relation. A paper by Douady and Lazard in the first volume of Inventiones deals with the case of a bundle of groups. Pradines generealises both of these to a topological or Lie groupoid G, an open subset W of G containing the identities of G, and a `local morphism' f: W \to H to Lie group H. There is a canonical local morphism and injection i: W \to MW where MW is an algebraically defined groupoid and such that any local morphism on W to a groupoid extends to a groupoid morphism on MW. The problem is to get a topology on MW so that it becomes a Lie groupoid and smooth local morphisms on W extend to smooth morphisms on MW. Pradines realised that the topology extension problem from i(W) to MW could be solved by holonomy methods, under certain conditions. So he had a construction of a Lie groupoid Hol(K,W) for a `locally Lie groupoid' (K,W). In the case K=MW, we find Hol(MW,W)=MW and so MW becomes a Lie groupoid (under the required holonomy conditions), and smooth local morphisms on W extend smoothly to MW. This seems a reasonable answer to the `globalisation of local morphisms'. The full details of these results are written up (with Pradines interest and approval) in theses of and papers with Mohammed Aof and with Osman Mucuk which I mentioned before on this list and available with others from http://www.bangor.ac.uk/~mas010/brownpr.html with publication details. What struck me about Pradines' results was that they were universal properties in differential topology, which seemed unusual, and probably extendible to other situations. The construction of Hol(K,W) is based on the idea of `iteration of local procedures no longer being local' and so is intuitively related to the applications suggested by Robert, and which can be found by web searches on `monodromy' and `holonomy' (which may get confused in the literature). In Pradines formulation, holonomy is a right adjoint and monodromy is a left adjoint. But this formulation in terms of germs and adjoints has yet to be written down in detail. It would be interesting to know if, where and how these formulations in terms of univeral properties can be applied, and related to common applications. (A paper `Towards 2-dimensional holonomy' with Ilhan Icen is to appear in Advances in Math.) I suppose one also has to ask about relations with descent and other ideas on local-to-global problems! So I agree with Robert that this work is more differential topology than category theory. Does that suggest work to be done in giving a more categorical formulation with wider applications? In another direction, there is presumably work for C^*-algebraists for the C^*-algebras on these more general holonomy and monodromy groupoids. (Connes' book on non-commutative geometry deals with the C^*-algebras of groupoids which are either equivalence relations or the holonomy groupoids of foliations, as I understand it.) Ronnie Brown ----- Original Message ----- From: "Robert L. Knighten" <Robert@knighten.org> To: <categories@mta.ca> Sent: Friday, August 23, 2002 1:59 AM Subject: categories: Categorical Monodromy? the the
web:
http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/93 /algtop93.html
projecteuclid.org/Dienst/Repository/1.0/Disseminate/euclid.dmj/1014136431/bo dy/pdfview
modular.fas.harvard.edu/papers/ants/kohel_stein.ps www.pacjmath.org/p/2002/205-2-2.pdf www.mccme.ru/ium/ium10/katzar.html
The first is the arichive of papers by Ronald Brown, a number of which discruss monodromy in the context of homotopy theory. Other areas where it appears include algebraic geometry, number theory, PDEs, . . .
-- Robert L. Knighten Robert@Knighten.org
25-Aug-2002 12:43:46 -0300,1136;000000000000-00000000