Local Homotopy Theory - Jardine, John F (Springer, 2015)
The subject of this monograph is the homotopy theory of diagrams of spaces, chain complexes, spectra, and generalized spectra, where the homotopy types are determined locally by a Grothendieck topology. The main components of the theory are the local homotopy theories of simplicial presheaves and simplicial sheaves, local stable homotopy theories, derived cate gories, and non-abelian cohomology theory. This book presents formal descriptions of the structures comprising these theories, and the links between them. Examples and sample calculations are provided, along with some commentary. The subject has broad applicability. It can be used to study presheaf and sheaf objects which are defined on the open subsets of a topological space, or on the open subschemes of a scheme, or on more exotic covers. Local homotopy theory is a foundational tool for motivic homotopy theory, and for the theory of topological modular forms in classical stable homotopy theory. As such, there are continuing applications of the theory in topology, geometry, and number theory. The applications and extensions of the subject comprise a large and expanding literature, in multiple subject areas...