Re: property_vs_structure

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

Dear George,
>
Thanks for your interesting response. I contend that "unramified coverings" is not an appropriate expression to describe those "coverings" not associated with a Galois theory in your sense since there is a specific meaning attached to it. I would simply call them "generalized coverings".  Such would be the unramified coverings or just the complete spreads. In fact also  the branched coverings but there is a difference here. For branched coverings, the motivation of it all,  a "branched Galois theory" already exists informally in the subject of knot groupoids. Moreover,  this example gives you the reason concerning coverings and Galois theories.  I devote one paragraph to each contention. 
>
1. Terminology "unramified". When R.H.Fox 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. 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". That is why I suggested "generalized coverings" - for both the ramified and the unramified which are not locally constant. >
2. Do we need a generalized Galois theory to deal with the branched coverings? Perhaps but, if so, it would not be a big generalization. Indeed,   the  "branched coverings", which are very important in topology, although they do not correspond directly to any Galois theory in your sense, they do through a biequivalence. Indeed for a topos E (say), 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 (Bunge-Niefield 2000, Funk 2000, Bunge-Lack 2003).  Now, C(E/Y) is in turn viewed as the knot groupoid G(K) of K. So, branched coverings branching over a knot is not an instance of coverings in your sense as they are not locally constant,  yet there is a Galois theory  associated with them, however, only through a non-trivial equivalence.  A strange but not unmanageable situation. 
> 
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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