property_vs_structure

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

Michael Shulman wrote:
>
> property = forgetful functor is full and faithful
> structure = forgetful functor is faithful
> property-like structure = forgetful functor is pseudomonic
>

On the thread "property"  "structure"  "property-like structure" and may be
some other etceteras.

I put on the table the following example to be analyzed:


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

Well, both are "properties" of a continuous function, but they are not of the
same kind.

in b) is hidden a structure, namely a trivialization structure associated to
an open cover of B.

If B is locally connected, then "covering map" behaves like a perfectly pure
property.

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).

have fun  !  e.d.


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