In various branches of mathematics one finds diverse notions of compactification, used for diverse purposes. Certainly one does not expect all instances of "compactification" to be specializations of one unique general notion. But perhaps there is something to be said from a more "taxonomic" perspective? That is, can we systematically categorize what are the chief distinctions to be drawn between different types of "compactification"?
Let's look at some examples. I would love to get some more examples to add to this list.
one-point compactification of locally compact spaces.
Stone-Cech compactification of completely regular spaces.
Bohr compactification of a topological group.
Wonderful compactification of a $G$-space.
Deligne-Mumford compactification of a moduli stack of curves.
- end compactification of a manifold.
- Various spacetime compactifications
I'm getting increasingly out of my depth as I go on, but let's list some
One needs a notion of "compact".
One identifies a class of "nice" spaces and canonical maps to "compact" spaces. Such maps should have "dense image" in an appropriate sense.
One is typically interested in cases where the canonical maps are "embeddings" in an appropriate sense.
One might try to compactify in a "maximal" or "minimal" way.
One may wish to have some interpretation of the new points as "ideal points" of the original space, e.g. "points at infinity". These might be equivalence classes of some kind of "line" in the old space for example.
In the case where one is compactifying some kind of moduli space, one likes to have a geometric interpretation of the new points one is adding, so that the compactification is also some kind of moduli space.
Sometimes one is interested in compactifying a broad class of spaces, and may want some kind of universal property.
Other times, one is compactifying one or a handful of particular space(s), and the emphasis is more on the geometric interpretation of the new points one is adding.
Question: Are there further commonalities between different notions of compactification? Are there further important distinctions to be drawn? To what extent is there a general theory of "compactification"?