Timeline for Is there a general theory of "compactification"?
Current License: CC BY-SA 4.0
13 events
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Oct 16, 2019 at 20:37 | comment | added | Ángel Valencia | I wish I thought the notion of "compactification" can be seen as a "nice" functor from a category (for example, all locally compact spaces) to a subcategory of "compact" objects (for example, all compact spaces), but I don't know if it works well in a general case... | |
Oct 15, 2019 at 21:33 | comment | added | André Henriques | 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). | |
Oct 15, 2019 at 12:53 | comment | added | Tim Campion | @TobyBartels Thanks, hemicompact spaces are indeed much more general than manifolds! I'm a little confused by the nlab description for arbitrary topological spaces, which I think you wrote judging by the revision history. I've dropped a note at the nforum. | |
Oct 15, 2019 at 11:54 | comment | added | Toby Bartels | 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). | |
Oct 15, 2019 at 8:46 | comment | added | Sebastian Goette | 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. | |
Oct 15, 2019 at 1:18 | comment | added | Robert Furber | 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). | |
Oct 14, 2019 at 22:54 | history | became hot network question | |||
Oct 14, 2019 at 20:51 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 61 characters in body
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Oct 14, 2019 at 17:00 | comment | added | user108998 | 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 | |
Oct 14, 2019 at 15:33 | answer | added | Andreas Blass | timeline score: 27 | |
Oct 14, 2019 at 15:15 | review | Close votes | |||
Oct 16, 2019 at 6:14 | |||||
Oct 14, 2019 at 15:00 | history | edited | Martin Sleziak |
I think that the (compactifications) tag might be suitable for this question
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Oct 14, 2019 at 14:48 | history | asked | Tim Campion | CC BY-SA 4.0 |