nLab motivic coarse space

Grothendieck topology on bornological coarse spaces

The category of bornological coarse spaces admits the structure of a site with an interval?. See §3 in Bunke and Engel for the relevant definitions.

Definition

The $\infty$-category of motivic coarse spaces is obtained from the $\infty$-category of $\infty$-sheaves on the site of bornological coarse spaces by performing the reflective localization with respect to the maps $I\times X\to X$ for any representable $X$, where $I=\{0,1\}$ is the interval object in bornological coarse spaces. (This step is formally identical to the one used to construct motivic spaces.) Next, we also localize with respect to morphisms of representables $\emptyset\to E$, where $E$ is a flasque bornological coarse space, as defined in Definition 3.19 of Bunke and Engel. Finally, we localize with respect to morphisms of presheaves of the form $colim_{U\in C}X_U\to X$, for any $C\in C$, where $X_U$ is obtained from $X$ using the same bornology and the coarse structure generated by a single element $U$.

References

• Ulrich Bunke, Alexander Engel,

  Homotopy theory with bornological coarse spaces, arXiv:1607.03657.