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I am following Clara Löh's Geometric Group Theory. An Introduction, and in remark 5.1.12, she defines the category QMet whose objects are metric spaces and whose morphisms are quasi-isometric imbeddings, identifying two morphisms if they are close.

Before reading this definition, I thought that morphisms would be defined as any maps $f:X\to Y$ such that for some constants $A,B$, we have $d(f(x),f(x'))\leq Ad(x,x')+B\;\forall x,x'\in X$, and then we identify morphisms if they are close (so for example, constant maps would be morphisms). This just seemed to be the natural generalization of the category of metric spaces. With this definition, the isomorphisms are still quasi-isometries. My question is, are there a lot of situations where the more restrictive definition is more useful?

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    $\begingroup$ 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.) $\endgroup$
    – YCor
    Commented Sep 19, 2022 at 4:46
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    $\begingroup$ 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. $\endgroup$
    – YCor
    Commented Sep 19, 2022 at 9:10
  • $\begingroup$ 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 $\endgroup$
    – Saúl RM
    Commented Sep 23, 2022 at 16:26
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    $\begingroup$ Actually ends is functorial under much more than quasi-isometric embeddings, namely proper large-scale Lipschitz maps. $\endgroup$
    – YCor
    Commented Sep 23, 2022 at 16:28
  • $\begingroup$ 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 $\endgroup$
    – Saúl RM
    Commented Sep 23, 2022 at 22:12

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