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.


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×XXI\times X\to X for any representable XX, where I={0,1}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 E\emptyset\to E, where EE 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 UCX UXcolim_{U\in C}X_U\to X, for any CCC\in C, where X UX_U is obtained from XX using the same bornology and the coarse structure generated by a single element UU.


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