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In the context of coarse spaces, a map between coarse spaces $f:X\to Y$ is called coarse if it is bornologous (it maps controlled sets to controlled sets), and proper, in the sense that preimages of bounded subsets are bounded. (A subset $B$ is bounded if $B\times B$ is controlled, or equivalently if $B\times\{x\}$ is for some point $x$.)

Why this last requirement? Why, for example, don't we want the constant map $\Bbb{R}\to\Bbb{Z}$ given by $x\mapsto 0$ to be a coarse map? What breaks down if we allow those?

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    $\begingroup$ It's a strange and unpractical convention, indeed. I encourage you not to include properness in the definition. $\endgroup$
    – YCor
    Commented Jan 15, 2020 at 18:47
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    $\begingroup$ I agree with @YCor. I am currently writing* a text taking a categorical perspective on the subject in which I omit the proneness property from the definition. $\endgroup$
    – Uri Bader
    Commented Jan 15, 2020 at 19:23
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    $\begingroup$ * "writing" in my comment above should be taken in the weak sense: this project is kept being postponed due to other administrative duties and more urgent mathematical projects. I am not sure this process is converging for me :( $\endgroup$
    – Uri Bader
    Commented Jan 15, 2020 at 19:23
  • $\begingroup$ @UriBader I'd love to read that once it's ready. (In the meantime, is there any reference which does not use this convention?) $\endgroup$
    – geodude
    Commented Jan 16, 2020 at 3:47
  • $\begingroup$ @geodude, none that I am aware of. $\endgroup$
    – Uri Bader
    Commented Jan 16, 2020 at 9:13

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