Homotopy Theory with Bornological Coarse Spaces

Providing a new approach to assembly maps, this book develops the foundations of coarse homotopy using the language of infinity categories. It introduces the category of bornological coarse spaces and the notion of a coarse homology theory, and further constructs the universal coarse homology theory. Hybrid structures are introduced as a tool to connect large-scale with small-scale geometry, and are then employed to describe the coarse motives of bornological coarse spaces of finite asymptotic dimension. The remainder of the book is devoted to the construction of examples of coarse homology theories, including an account of the coarsification of locally finite homology theories and of coarse K-theory. Thereby it develops background material about locally finite homology theories and C*-categories. The book is intended for advanced graduate students and researchers who want to learn about the homotopy-theoretical aspects of large scale geometry via the theory of infinity categories.

- Introduction. - Part I Motivic Coarse Spaces and Spectra. - Bornological Coarse Spaces. - Motivic Coarse Spaces. - Motivic Coarse Spectra. - Merging Coarse and Uniform Structures. - Part II Coarse and Locally Finite Homology Theories. - Locally Finite Homology Theories and Coarsification. - Coarse K-Homology.