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.
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