nLab bornological coarse space

Definition

A bornological coarse space is a set $X$ equipped with a bornology $B$ and a coarse structure $C$ that are compatible: controlled thickenings of bounded subsets are bounded.

Here a U-controlled thickening ($U\subset X\times X$) of a subset $B$ of $X$ is the projection of $X\times B\cap U$ onto its first factor.

A morphism of bornological coarse spaces is a map $f$ of sets that is controlled (for every $U\in C$ we have $(f\times f)(U)\in C'$) and proper (for every $B'\in B'$ we have $f^{-1}(B')\in B$).

Bornological coarse spaces can be used as a site with an interval? to construct motivic coarse spaces.

References

• Ulrich Bunke, Alexander Engel,

  Homotopy theory with bornological coarse spaces, arXiv:1607.03657.