RE: property_vs_structure

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

Dear Eduardo,
Topological spaces or toposes, it is the same question. A space is locally connected iff its topos of sheaaves is locally connected. 
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.  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?  
This is all I meant. I was not disputing a well known fact about covering spaces of locally connected spaces (or of toposes, for that matter).
Best regards,Marta



> Date: Thu, 7 Oct 2010 21:40:54 -0300
> From: edubuc@dm.uba.ar
> To: marta.bunge@mcgill.ca
> CC: categories@mta.ca
> Subject: Re: property_vs_structure
> 
> I am talking about topological spaces
> 
> Marta Bunge wrote:
>> 
>> 
>> 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.
>> 
>>> Date: Wed, 6 Oct 2010 09:34:51 -0300
>>> From: edubuc@dm.uba.ar
>>> To: edubuc@dm.uba.ar
>>> CC: categories@mta.ca
>>> Subject: categories: errata
>>>
>>>
>>>>
>>>> The difference is only manifest when the space B is not locally
>>>> connected. In
>>>> this case we may have homeomorphisms from X to X over B which do not
>>>> preserve
>>>> this structure (Spanier, Algebraic Topology).
>>>>
>>>>
>>>
>>> is not quite it should be,
>>>
>>> there is a clear notion of isomorphism of trivialization structure, 
>> and a same
>>> space X over B may have non isomorphic structures. Alternatively, a 
>> continuous
>>> function over B does not necessarily preserve the trivialization 
>> structures.
>>>
>>> however, if B is locally connected, trivialization structures are 
>> like a pure
>>> property of X.
>>>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]