Timeline for Is there a universal property characterizing the category of compact Hausdorff spaces?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Aug 22, 2023 at 20:38 | vote | accept | Georg Lehner | ||
| Jan 29, 2023 at 15:58 | comment | added | Alex Kruckman | It's Stone-Čech (for Marshall Stone and Eduard Čech), not stone Czech. | |
| Jan 29, 2023 at 2:47 | answer | added | Martin Brandenburg | timeline score: 8 | |
| Jan 29, 2023 at 2:42 | comment | added | Martin Brandenburg | @Tyrone This is not a comment, it is a full answer. | |
| Jan 26, 2023 at 21:11 | history | became hot network question | |||
| Jan 26, 2023 at 20:54 | answer | added | Simon Henry | timeline score: 20 | |
| Jan 26, 2023 at 16:14 | comment | added | Tyrone | See V. Marra and L. Reggio, A Characterisation of the Category of Compact Hausdorff Spaces, Theory App. Cat. 35, (2020), pp.1871–1906 (arxiv version). The authors show that $CHaus$ is, up to equivalence, "the unique non-trivial pretopos which is well-pointed, filtral and admits all set-indexed copowers of its terminal object". Refer to $\S4$ of the paper for their notion of filtrality. | |
| Jan 26, 2023 at 15:24 | comment | added | Max New | Isn't the fact that it is the category of algebras for the ultrafilter monad already a universal property? | |
| Jan 26, 2023 at 14:51 | history | edited | Paul Taylor |
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| Jan 26, 2023 at 14:51 | answer | added | Paul Taylor | timeline score: 15 | |
| Jan 26, 2023 at 13:42 | comment | added | Robert Furber | Compact Hausdorff spaces also have a "minimal" cover by a Stonean space, which is obtained by taking the Stone space of the complete Boolean algebra of regular open sets (a construction due to Andrew Gleason). | |
| Jan 26, 2023 at 13:41 | comment | added | Robert Furber | $\mathbf{Prof}^{\mathrm{op}}$ is equivalent to the category of Boolean algebras and therefore locally $\aleph_0$-presentable. The category $\mathbf{CHaus}^{\mathrm{op}}$ is not locally $\aleph_0$-presentable, but it is locally $\aleph_1$-presentable (and in fact monadic over $\mathbf{Set}$) because it is equivalent to the category of commutative unital C$^*$-algebras. This can be stated as $\mathbf{CHaus}$ is the free completion of the compact metric spaces (equivalently, the closed subspaces of $[0,1]^\mathbb{N}$). Is this the kind of thing you're looking for? | |
| Jan 26, 2023 at 12:17 | history | asked | Georg Lehner | CC BY-SA 4.0 |