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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:

Algebraic Geometry:

Differential Geometry:

Mathematical Physics:

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"?

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    $\begingroup$ I think in more geom contexts (ie not point set topology) a fairly neat way to "define" the boundary at infinity would be as a pro-object in the appropriate category, ie the formal diagram lim (X\K) over compact subspaces K. (eg the hemicompact definition I just learned from one of ur links is a srequirement on the indexing category for this diagram). Presumably this says something about compactifications. I think I read this in a recent paper of toen and pantev $\endgroup$ – EBz Oct 14 at 17:00
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    $\begingroup$ This has come up before, but the Bohr compactification is not a compactification in one commonly used sense, because the universal map from an abelian group to its Bohr compactification is not a topological embedding (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). $\endgroup$ – Robert Furber Oct 15 at 1:18
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    $\begingroup$ Some compacitifactions try to keep some of the geometry of the space they compactify. For example, a conformal compactification tries to keep the conformal structure on the interior. In nice examples, it produces a manifold with boundary or with corners. The boundary at infinity of 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. $\endgroup$ – Sebastian Goette Oct 15 at 8:46
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    $\begingroup$ The end compactification is for a lot more than just manifolds. It's a point-set concept that works for any hemicompact space (and really any space at all, but as with the Stone–Čech compactification, I'm not sure if it's still a compactification outside of its original context). $\endgroup$ – Toby Bartels Oct 15 at 11:54
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    $\begingroup$ Dual question: is there a general theory of "making things discrete" [think: find an epimorphism from a set]? Examples: underlying set of a manifold; set of $k$-rational points of a variety; underlying vector space of a Banach space... Once again, if you want the result of this operation to not be too huge, you'll need to make (arbitrary) choices (e.g. the choice of a dense countable set inside a topological space). $\endgroup$ – André Henriques Oct 15 at 21:33
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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.

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    $\begingroup$ To me it seems that Stone–Čech has to be big because it tries to be universal. On the other hand, one-point compactifications and compactifications in algebraic geometry try to be nice spaces. $\endgroup$ – R. van Dobben de Bruyn Oct 14 at 16:03
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    $\begingroup$ In part this is because the universal objects tend not to exist within algebraic geometry. For instance the universal cover of a complex variety tends to be transcendental in nature and outside the algebraic geometry "universe"; instead one has to take formal inverse limits of finite extensions to work with things like the etale fundamental group as a substitute. Similarly any Stone-Cech type "universal blowup" of an algebraic singularity would be extremely non-algebraic in nature, so one works with one or more finitely complicated partial blowups instead. $\endgroup$ – Terry Tao Oct 14 at 23:05
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    $\begingroup$ @R.vanDobbendeBruyn That is a common line of thinking, based on familiar examples such as $\mathbb{N}$ and $\mathbb{R}$. But consider the fact that if you give $\aleph_1$ its order topology, its Stone-Čech compactification is the same as its 1-point compactification. $\endgroup$ – Robert Furber Oct 15 at 1:06
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    $\begingroup$ @RobertFurber That only holds because that ordinal space is “almost compact” to begin with, in quite a strong sense : it has a unique uniformity inducing its topology. $\endgroup$ – Henno Brandsma Oct 15 at 10:32

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