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.
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