Re: property_vs_structure

[Marta Bunge <marta.bunge@mcgill.ca>]

Dear George,
>
Thanks for your interesting response. Let me just comment here on your "addition 1" below. It is my contention that "unramified coverings" is not an  appropriate expression to describe those "coverings" not associated with a  Galois theory in your sense, as first, there is a specific meaning attached to it, and secondly, a "branched Galois theory" already exists informally in the subject of knot groupoids. One may possibly generalize your  Galois theories to include these phenomena. I devote one paragraph to each  contention. 

>
1. When R.H.Fox ("Covering spaces with singularities", R.H. Fox et al, editors, Algebraic Geometry and Topology: A Symposium in honor of S. Lefschetz, Princeton University Press, 1957, 243-257) introduced spreads and their completions, he had in mind what the title of his paper says,  that is, "coverings with singularities" so, not the traditional locally constant coverings. There could be "ramifications", or branchings over points of the base. But no folds. Specifically, he was thinking of branched coverings  (branching over a knot in the base) as the spread completions of locally constant coverings, in which branching points were added to the domain space. This is what led him to define a notion of spread, and then perform a completion process leading to another spread singled out among all such corresponding to a given cosheaf on the base space. The branched coverings, and more generally the complete spreads of which they are  the motivating example, are "ramified". Now, add the condition that the complete spread (e.g. a branched covering) be a local homeomorphism. This  does not force it to be locally constant, as we know, but it cannot then have ramifications. Hence the expression "unramified coverings". 
>
2. As I said in my last posting, the  "branched coverings", which are  very important in topology, yet do not correspond to any Galois theory in your sense, should correspond to a "generalized Galois theory" or to a "branched Galois theory". To support my contention, note that, in (M. Bunge and S. Lack, van Kampen theorems for toposes, Advances in Mathematics 179/2 (2003) 291-317), we obtain, as an application of the van Kampen theorems we prove therein,  a connection with the use of the automorphism group of a (universal) branched covering in the calculation of knot groups, as advocated by Fox. In particular, and the point I am making here, the expression "unramified coverings" does not describe them accurately, as there may be ramifications. For a topos E, there is a biequivalence of the 2-categories of branched coverings of E branching over an object  Y (the latter thought of as the complement of a knot K)  on the one hand, and that of all locally constant coverings of the slice topos E/Y on the other. The latter may in turn be viewed as the fundamental groupoid of E/Y, or as the knot groupoid G(K) of K. There is a "Galois theory" there not associated with coverings in your sense, that is, with locally constant coverings. 

>
All of this requires further investigation, for which I will have no time  possibly until December, due to my trip to Buenos Aires. 
>

With best regards,
>
Marta

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Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics 
McGill UniversityBurnside Hall, Office 1005
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marta.bunge@mcgill.ca 
http://www.math.mcgill.ca/~bunge/
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