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:**

one-point compactification of locally compact spaces.

Stone-Cech compactification of completely regular spaces.

Bohr compactification of a topological group.

**Algebraic Geometry:**

Wonderful compactification of a $G$-space.

Deligne-Mumford compactification of a moduli stack of curves.

**Differential Geometry:**

- end compactification of a manifold.

**Mathematical Physics:**

- Various spacetime compactifications

I'm getting increasingly out of my depth as I go on, but let's list some

**Commonalities:**

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:**

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"?

i.e.the original topology does not agree with the subspace topology on the image). Similarly, the restriction of the Stone-Čech compactification to completely regular spaces is not a necessary part of the construction, but arises because doing so makes the universal mapping an embedding (a space is completely regular iff it is embeddable in a compact Hausdorff space). $\endgroup$conformal compactificationtries to keep the conformal structure on the interior. In nice examples, it produces a manifold with boundary or with corners. Theboundary at infinityof a Cartan-Hadamard space can be defined so that it keeps track of directions of geodesics. These are examples of the very last bullet in the list. $\endgroup$2more comments