Timeline for Concrete topological objects and notions in the category of locales
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 10, 2022 at 12:26 | comment | added | Tyrone | Some references can be found at nlab. | |
| Jun 10, 2022 at 11:53 | comment | added | Zhen Lin | The category of locales has products, and the "forgetful" functor $\textbf{Top} \to \textbf{Loc}$ preserves all colimits, the terminal object and also binary products where at least one of the factors is locally compact Hausdorff. The "forgetful" functor is also fully faithful on the full subcategory of Hausdorff spaces. Thus, as Maxime says, if you want to focus on the homotopy theory of CW-complexes or other pleasant spaces then there is no theoretical obstruction to using locales. | |
| Jun 10, 2022 at 8:14 | comment | added | Maxime Ramzi | To some extent this is related to shape theory - in an arbitrary topos $X$ (in particular the sheaf topos of a locale) you can describe cohomology with constant coefficients (which, in the case of nice spaces, recovers singular cohomology); you can also describe a (pro-)fundamental groupoid and in fact a (pro-)fundamental $\infty$-groupoid. I don't think looking for CW-complexes and ordinary spaces such as $S^1$ leads to interesting things in the sense that these are sober spaces and so are completely characterized by their locales | |
| S Jun 10, 2022 at 7:05 | review | First questions | |||
| Jun 10, 2022 at 7:42 | |||||
| S Jun 10, 2022 at 7:05 | history | asked | user1892304 | CC BY-SA 4.0 |