Timeline for Definition of the category QMet of metric spaces and quasi-isometries
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Sep 23, 2022 at 23:00 | comment | added | Saúl RM | To be honest I hadn't read the definition of large scale Lipschitz and I assumed you meant that this could be extended to coarse maps and not only quasi-isometries. But now I understand the comment. In fact Löh's book also uses proper large-scale Lipschitz maps in the chapter about ends (for example to define quasi-rays and the Gromov boundary) | |
| Sep 23, 2022 at 22:16 | comment | added | YCor | I don't get your last comment: every proper large-scale Lipschitz map induces a continuous map between space of ends, in a functorial way. There are many such maps that are not QI-embeddings. | |
| Sep 23, 2022 at 22:12 | comment | added | Saúl RM | Oh yes, sorry, I meant that from the morphisms I said above the only ones which seem to make boundaries work are quasi-isometric imbeddings, but yeah it makes sense that we can generalize quasi-isometric maps to something "not linear". Btw thanks for all the answers, including the one from MSE a few days ago | |
| Sep 23, 2022 at 16:28 | comment | added | YCor | Actually ends is functorial under much more than quasi-isometric embeddings, namely proper large-scale Lipschitz maps. | |
| Sep 23, 2022 at 16:26 | comment | added | Saúl RM | After reading parts of Chapter 8, I think one of the main reasons she introduces the more restrictive category is to deal with ends: for example, she thinks of the notion of boundary as a functor from QMet to the category of topological spaces, and she defines rays as morphisms from $[0,\infty)$ to a space. This only makes sense if we use the category whose morphisms are quasi-isometric imbeddings | |
| Sep 19, 2022 at 9:10 | comment | added | YCor | Oops, here the author considers all QI embeddings as morphisms, so it's not just restricting to isomorphisms. I'm not sure I know a purely categorical characterization of QI embeddings within the large scale category. Still, let's stay it's natural to define the large scale category and then consider various subcategories. Categories are notably useful to define functors, and sometimes functors are defined only on some given subcategory. There are also larger categories that are useful in this context, such as considering all coarse maps, etc. | |
| Sep 19, 2022 at 4:46 | comment | added | YCor | I also believe the definition you're providing is more useful: moreover it provides by itself a natural definition of quasi-isometry (= maps inducing isomorphism in the large-scale category). Besides, whenever you have a category $C$, you get a new category $C'$ by taking the same objects and restricting to isomorphisms. So $C'$ is derived from $C$, but $C$ usually contains much more information. (The large-scale category appears e.g. Def. 3.A.11 in this book.) | |
| Sep 19, 2022 at 4:09 | history | asked | Saúl RM | CC BY-SA 4.0 |