Re: property_vs_structure

["Eduardo J. Dubuc" <edubuc@dm.uba.ar>]

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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