Timeline for Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)?
Current License: CC BY-SA 2.5
13 events
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| Mar 23, 2023 at 18:23 | comment | added | user19232801 | @AndréHenriques I want to emphasize that condensed sets (i.e. sheaves on compact Hausdorff spaces) already appeared in the comment chain! | |
| Nov 8, 2021 at 15:29 | comment | added | André Henriques | @CharlesRezk. "these improved categories of topological spaces (CG, CGH, CGWH, etc.) are trying to do is to make a version of Top which is as close to having all the properties of a topos as is possible". With the advent of condensed sets, we now have a version of Top which is a topos on the nose (with the caveat that there's no set of generators). I guess this means that we could go back and rewrite all the old topology texts with the words 'condensed set' in place of 'CG(W)H space', and all the proofs will work verbatim. | |
| May 18, 2011 at 16:04 | comment | added | David Carchedi | @Tyler: I believe what you're looking for is the following: Let $j:CH \hookrightarrow Top$ be the inclusion of compact Hausdorff spaces into topological spaces. It induces a geometric morphism $\left(j_*,j^*\right)$ between the topoi of sheaves $$Sh\left(CH\right) \to Sh\left(Top\right).$$ The category of compactly generated spaces is equivalent l to the essential image of the restriction of $j^*$ to representable sheaves (i.e.topological spaces). If $Top$ is replaced by Hausdorff spaces or weakly Hausdorff spaces, the analogous statement is also true. | |
| Feb 16, 2011 at 17:16 | vote | accept | André Henriques | ||
| Nov 30, 2010 at 4:02 | comment | added | Tyler Lawson | @Charles: I remember some comment made (I think by MJH) about the "convenient category" being either equal to, or related to, a category of sheaves on compact spaces. However, I've never been able to reconstruct the proper statement - is it familiar to you? | |
| Nov 30, 2010 at 1:55 | comment | added | Todd Trimble | Yes, it seems that the chief merit Spanier's quasitopological spaces have is that they are locally cartesian closed. Also worthy of mention is Johnstone's topological topos. | |
| Nov 30, 2010 at 1:52 | comment | added | Todd Trimble | Does "quotient" invariably refer to quotient in Top? I would have been inclined to interpret it as a coequalizer of the two projections off an equivalence relation (as interpreted in whatever category), thus a special coequalizer. | |
| Nov 30, 2010 at 1:25 | comment | added | Charles Rezk | It has always struck me that what these "improved" categories of topological spaces (CG, CGH, CGWH, etc.) are trying to do is to make a version of Top which is as close to having all the properties of a topos as is possible. (And this is why some people prefer to work with simplicial sets instead of spaces; simplicial sets is already a topos!) | |
| Nov 30, 2010 at 1:22 | comment | added | Charles Rezk | @ Todd, yes that would be amazing. However, I don't think all coequalizers are quotient maps in CGWH (cf Dan Ramras's comment under Harry's answer), so I don't think pullbacks of coequalizers are necessarily coequalizers. | |
| Nov 29, 2010 at 23:30 | comment | added | Todd Trimble | That the pullback of a quotient is a quotient sounds strangely close to suggesting that CGWH is locally cartesian closed (and closer still if the pullback of a coequalizer is a coequalizer), and this would be amazing to me. Charles, does Lewis's thesis (or any of the other references) address this issue? | |
| Nov 29, 2010 at 21:40 | history | edited | Charles Rezk | CC BY-SA 2.5 |
Correction about regular epis
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| Nov 29, 2010 at 21:35 | history | edited | Charles Rezk | CC BY-SA 2.5 |
Added note on McCord's paper
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| Nov 29, 2010 at 21:27 | history | answered | Charles Rezk | CC BY-SA 2.5 |