I've heard about things in math that have been "completely classified," like finite simple groups, and other things that are not completely classified, like topological spaces. Does "classification" in this sense have a generally accepted and precise meaning?
By "precise," I'm hoping it could clarify questions like the following:
Does "complete classification" of some objects necessarily mean that we can write an algorithm that determines whether two input objects are isomorphic, where the algorithm completes after a finite number of steps? That's what this answer suggests, but this seems overly restrictive. Suppose I have a finite list of properties such that iff $A$ and $B$ agree on all properties, then $A$ and $B$ are isomorphic. Suppose further that there is no algorithm that can, in a finite number of steps, always determine whether these properties agree for any two input objects. Is this a still classification?
Suppose I have a finite list of properties such that iff $A$ and $B$ agree on all properties, then $A$ and $B$ are isomorphic. But suppose the "value" of one such property takes on a type of object that is not completely classified. For example, suppose I have such a list of properties for classifying manifolds, and that one of these properties is a certain Banach space associated with the manifold. I.e., if two manifolds have the same (or isomorphic) associated Banach space, and also agree in all of the other properties on my list, then the manifolds are isomorphic. Banach spaces, however, are not completely classified. Does this mean I would still not have completely classified manifolds?
- Suppose I don't have a list of properties that uniquely determine each object (up to isomorphism), but instead have a list of properties that still somehow provide a nice organization scheme. Is this ever considered a classification?