Homotopy theory with bornological coarse spaces

We propose an axiomatic characterization of coarse homology theories defined on the category of bornological coarse spaces. We construct a category of motivic coarse spectra. Our focus is the classification of coarse homology theories and the construction of examples. We show that if a transformation between coarse homology theories induces an equivalence on all discrete bornological coarse spaces, then it is an equivalence on bornological coarse spaces of finite asymptotic dimension. The example of coarse K-homology will be discussed in detail.Comment: 220 pages (complete revision

Coarse cohomology theories

We propose the notion of a coarse cohomology theory and study the examples of coarse ordinary cohomology, coarse stable cohomotopy and coarse cohomology theories obtained by dualizing coarse homology theories. Our investigations of coarse stable cohomotopy lead to a solution of J. R. Klein's conjecture that the dualizing spectrum of a group is a coarse invariant. We further investigate coarse cohomological $K$-theory functors and explain why (an adaption of) the functor of Emerson--Meyer does not seem to fit into our setting.Comment: 55 page