Is there a general theory of "compactification"? 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.
Topology:


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*one-point compactification of locally compact spaces.

*Stone-Cech compactification of completely regular spaces.

*Bohr compactification of a topological group.
Algebraic Geometry:


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*Wonderful compactification of a $G$-space.

*Deligne-Mumford compactification of a moduli stack of curves.
Differential Geometry:


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*end compactification of a manifold.


Mathematical Physics:


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*Various spacetime compactifications
I'm getting increasingly out of my depth as I go on, but let's list some
Commonalities:


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*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.
Distinctions:


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*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"?
 A: There's a distinction that I find striking but don't know how to formalize usefully or how to evaluate its importance: In algebraic geometry, moduli spaces get compactified, and this involves adding a relatively small set to the original space. Roughly speaking, the original space parametrizes some nice objects, and the compactification adds points "at infinity" parametrizing some sort of degenerations of those nice objects. Typically, the part at infinity has lower dimension than the original space. In contrast, in "my world" of ultrafilters and Stone-Cech (or similar) compactifications, the part at infinity tends to be far bigger than the original space. A countable discrete space and the real line both have Stone-Cech compactifications of cardinality $2^c$ where $c$ is the cardinal of the continuum. 
And it's not just a matter of the size of the part at infinity; it's also a matter of the intuitive interpretation of those extra points. It's difficult for me to imagine a non-principal ultrafilter on $\mathbb N$ as a "degenerate" natural number.
