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