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