RE: property_vs_structure

[Marta Bunge <marta.bunge@mcgill.ca>]

Dear Eduardo,>In a  previous message of mine and  in response to your  questions >
>> What do we mean by structure ?, and, what do we mean by property  ?
>
 I wrote>
>> On a given data, a structure is additional data on it such 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).>
to which you replied>  
>> 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.>
Notice that I specified "in the model theoretic sense". In that context, the notions of a structure and of a morphism of structures are defined and it is that sense that I meant them. They are perfectly meaningful. 
>
You added> 
>> 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
> **********
>
I do not "subscribe" to these notions. I simply use the well-known notions from first-order logic and model theory as I said earlier. I am aware of the difference between the locally connected  and the general case concerning covering projections and it is not the mathematics that I was disputing.  But I still have trouble following your analysis of this situation as was your purpose. The difference is that, whereas you consider the notion of a covering projection to be a property of a continuous map p from X to B which, in the general (non locally connected case) is a "hidden structure", I view the notion of a covering projection to be a structure in the first place and, in the specific locally connected case, one that may be equivalently reduced to a property. Every property is a structure, but not  every structure is  reducible to a property. 

>
All the best,  Marta



  		 	   		  

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