Dear Eduardo,
You ask
What do we mean by structure ?, and, what do we mean by property ?
On a given data, a structure is additional data on it that, if it exists, it is not necessarily unique up to isomorphism. A property is additional data such that, if it exists, it is unique up to isomorphism (in the model theoretic sense).
You also write
Finally, I still do not understand what do you mean (in your first mail) by:>>
Even in the locally connected case there are several non isomorphic trivialization structures. The difference is that, in that case, there is a canonical one.
Since in this case all trivialization structures ARE isomorphic!.
(if U and V are neighborhoods of b evenly covered, then the structures are isomorphic in a connected W contained in the intersection)
Precisely, it comes down to what definition of covering space one adopts. If the general definition is adopted then, even in the locally connected case, it is a structure, as any two trivialization structures given by U, V, need not be isomorphic except on a connected W contained in the intersection. If, on the other hand, the specifi definition is adopted, where canonical neighborhoods (U open, connected, and each connected component of the inverse image of U under p in X is mapped homeomorphically onto U) are given as part of the structure, then the structure is a property.
I think that I have nothing else to say on his matter.
Best,Marta
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Date: Fri, 8 Oct 2010 18:53:31 -0300 From: edubuc@dm.uba.ar To: marta.bunge@mcgill.ca CC: categories@mta.ca Subject: Re: property_vs_structure
Marta Bunge wrote:
Dear Eduardo, Topological spaces or toposes, it is the same question. A space is locally connected iff its topos of sheaaves is locally connected.
Of course, it is only that I wanted to focus in topological spaces to fix the ideas and so that the following two definitions can be compared.
ALSO, I had FORGOTTEN to say that in definition b) the V's are DISJOINT.
******** Let f: X --> B a continuous function of topological spaces: [assume surjective to simplify, and if b \in B, write X_b for the fiber X_b = f-1(b)].
Then, we have the two familiar definitions a) and b):
f is "fefesse" if given b \in B, then
a) for each x \in X_b, there is U, b \in U, such that
b) there is U, b \in U, such that for each x \in X_b,
there is V, x \in V, and f|V : V --> U homeo.
(the non commuting quantifiers again !)
a) fefesse = local homeomorphism
b) fefesse = covering map **********
In my view, the question of whether the notion of a covering space is a structure or a property depends on the definition of covering space that one adopts. If the definition is made for arbitrary spaces (as in Spanier, whom you quote), where a continuous map p from X to B is said to be a covering projection if each point of X has an open neighborhood U evenly covered by p, then covering space is a structure, no matter what the nature of the base space is.
Well, for locally connected space B (or any locally connected topos as you pointed out), the forgetful functor into the topos of etale spaces over B is full and faithful, and for X over B, there is only one structure (up to isomorphism of structures).
I wanted this to be considered under the analysis:
*************** Michael Shulman wrote:
property = forgetful functor is full and faithful structure = forgetful functor is faithful property-like structure = forgetful functor is pseudomonic
***************
You see, with this criteria (property = forgetful functor is full and faithful) covering space is a property, something you do not think it is. I am not saying who is right, just putting in evidence that it is a matter not settled yet. May be full and faithfulness of the forgetful functor is not enough to call a covering space to be a property of a continuous map ?
It so happens that, in the case of a locally connected space B, an alternative definition of a covering space can be given (as in R. Brown, Topology and groupoids) that refers directly to canonical neighborhoods of points of X (U open, connected, and each connected component of the inverse image of U under p in X is mapped homeomorphically onto U) and, with this definition, covering space is indeed a property. So, in the locally connected case, the structure of covering space can be equivalently replaced by a property - but I believe that it is still a structure before those canonical choices are made. Can a structure be equivalent to a property, yet not be a property?.
Well, interesting question, but first we have to settle:
What do we mean by structure ?, and, what do we mean by property ?.
Finally, I still do not understand what do you mean (in your first mail) by:
Even in the locally connected case there are several non isomorphic trivialization structures. The difference is that, in that case, there is a canonical one.
Since in this case all trivialization structures ARE isomorphic!.
(if U and V are neighborhoods of b evenly covered, then the structures are isomorphic in a connected W contained in the intersection)
best e.d.
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Marta Bunge