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There is a large popularity of the topic of coarse geometry aka coarse geometry which is kind of topological subject dual to usual topology. Say, if we have a metric space, topological structure is about continuity at small scale, while the coarse structure is about a large scale structure. The subject has largely being motivated by strong results of Gromov on hyperbolic groups, asymptotic dimension and so on. One can look at the large scale structure in a metric space, but one can also, take a more general, axiomatic approach via a coarse structure. The subject has also interfaces with operator algebras, Novikov conjecture, Baum-Connes conjecture and so on. Some of the leading researchers are Dranishnikov, John Roe, Higson etc. Regarding that the subject is so similar and parallel to topology (say one has cohomological asymptotic dimension wth smilar properties…) and the homotopic part of the topology can be expressed in terms of inifinity topos, and much of the usual topology like sheaves in terms of usual topos theory. Are there petit topoi and infinity gros topoi for coarse geometry or variants thereof ? Is anybody working on it ?
@Zoran: I expect you had seen this, but there is a very recent paper by Roe and Siegel: J . K-Theory, 12 (2013), 213–234 doi:10.1017/is013006016jkt233 that may be of interest. I have not looked at it in detail.
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