Timeline for Regular epi- and mono-morphisms for locally compact (Hausdorff) groups
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 29, 2019 at 15:18 | comment | added | Yemon Choi | @TimCampion One needs to be careful waving the phrase "Gelfand duality" around when dealing with locally compact groups and continuous homomorphisms between them, especially because many of the interesting homomorphisms are not proper maps. | |
| Oct 28, 2019 at 18:01 | comment | added | Tim Campion | Is the category of locally compact groups cocomplete (I'm not sure, but my casual understanding of Gelfand duality suggests that it is)? If so, then note that in a finitely complete and cocomplete category, a monomorphism $i: G' \to G$ is regular iff it is the equalizer of the two maps $G' \rightrightarrows G \ast_{G'} G$, where $G \ast_{G'} G$ denotes the pushout of $i: G' \to G$ along itself (the "cokernel pair" of $i: G' \to G$). So if one can understand such pushouts, one will have a criterion. | |
| Oct 28, 2019 at 14:35 | comment | added | Matthew Daws | @LSpice: Can I put arrows above and below $\underset{h}{\overset{g}{\rightrightarrows}}$ Ah, yes, that works. Thanks! | |
| Oct 28, 2019 at 12:48 | comment | added | LSpice |
In terms of diagrams, do you want just $\rightrightarrows$ $\rightrightarrows$? MathJax also supports AMScd (math.meta.stackexchange.com/questions/2324/…).
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| Oct 28, 2019 at 12:37 | comment | added | YCor | I also doubt of a complete description of epimorphisms in the category of locally compact groups. In a connected semisimple Lie group $G$, I think the inclusion of every Zariski-dense closed subgroup is an epimorphism of locally compact groups. | |
| Oct 28, 2019 at 12:19 | history | edited | YCor | CC BY-SA 4.0 |
fixed typo, added intials
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| Oct 28, 2019 at 12:16 | comment | added | YCor | The second part is correct. No, it's not the same as for discrete groups, since these are only quotients by closed normal subgroups (when you work in the category of Hausdorff topological groups or its locally compact subcategory). If you work with $G,H$ Hausdorff but in the category of all topological groups, this may modify the coequalizer. | |
| Oct 28, 2019 at 11:43 | history | asked | Matthew Daws | CC BY-SA 4.0 |