Re: property_vs_structure

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

Marta Bunge wrote:
> Dear Eduardo,>>You ask>>>> What do we mean by structure ?, and, what do we
> mean by property ?.>>On a given data, a structure is additional data on it
> that, if it exists, it is not necessarily unique up to isomorphism. A
> property is additional data such that, if it exists, it is unique up to
> isomorphism (in the model theoretic sense).

Well, here it is necessary first to establish what do we mean by
"isomorphism". To do this we need a way to compare the structures, that is, we
have to define morphism of structures (see below). Without this, the above is
meaningless.

People discussing structure vs property were giving examples where all this
was clear and straightforward (invertibility in a monoid, neutral element in a
semigroup, etc). My original purpose when I wrote my first mail was to
consider a less trivial example testing the following  "definitions", that I
see you subscribe above at least in what it concerns "structure" and "property":

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

>> You also write>>>> 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 interesection)
>>>
> Precisely, it comes down to what definition of covering space one adopts.
> If the general definition is adopted then, even in the locally connected
> case, it is a structure, as any two trivialization structures given by U,
> V, need not be isomorphic except on a connected W contained in the
> intersection. If, on the other hand, the specific definition is adopted,
> where canonical neighborhoods (U open, connected, and each connected
> component of the inverse image of U under p in X is mapped homeomorphically
> onto U) are given as part of the structure, then the structure is a
> property. >> I think that I have nothing else to say on his
> matter.>>Best,Marta

Let us analyze carefully: take definition a): "open neighborhood U of b evenly
covered", say in X and in Y (over B). Given two structures (in the same U),
the definition of morphism is clear. It is a continuous function from X to Y
over B together with some extra data. When U is not connected, a function over
B does not necessarily carries the extra data, and the forgetful functor is
not full. There are non isomorphic structures on the same X. It is not a
property. When U is connected (no loss of generality in the locally connected
case), any continuous function over B carries automatically this extra data.
The forgetful functor is full, and for a given X, there is only one structure
up to isomorphisms. It is a property according to  definition (**) above.

All this if we have a fixed U (or a fixed cover of B). But we need all
covering projections, trivialized over all possible covers.

Now, what happens if we have an structure in a different open neighborhood V
of b. How we define morphism ?. Well, we take the restriction of the
structures to an open W contained in U and in V, and we are in the previous
case. In this way, when U, V and W are all connected, any continuous function
over B carries automatically this extra data. The forgetful functor is full,
and for a given X, there is only one structure up to isomorphisms.

It seems to me that, once you define the meaning of isomorphism between
structures, in the locally connected case for a given X there is only one
structure up to isomorphism.
Consider what  you call "the specific definition" b) (U open, connected, and
each connected component of the inverse image of U under p in X is mapped
homeomorphically onto U) which makes sense only for the locally connected case
(it use connected components). Well, both definitions are  equivalent, and
they are both "properties" according to the definition of property (**) quoted
above.

Classically, covering projections were treated as spaces with a certain
property, and so defined. And everything was fine since immediately after the
definition, it is assumed local connectivity once they start to develop the
subject. In shape theory they had to deal with non locally connected spaces,
and they run into trouble. There was a "hidden structure" which can not be
ignored in the non locally connected case.

It was this hidden structure that I put into consideration in order to analyze
if the concepts of property and  structure given in (**) are or are not fine
enough.

  From my previous mails:
*****
                  local homeomorphism   covering map

Both are "properties" of a continuous function, but they are not of the same kind.
in "covering map" there is hidden a structure (not considered classically),
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.
*****
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 ?
******

best to all   e.d.

> ----------------------------------------
>> Date: Fri, 8 Oct 2010 18:53:31 -0300 From: edubuc@dm.uba.ar To:
>> marta.bunge@mcgill.ca CC: categories@mta.ca Subject: Re:
>> property_vs_structure
>>
>>
>>
>> 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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