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