# Local Homotopy Theory

Part of the Springer Monographs in Mathematics book series (SMM)

Part of the Springer Monographs in Mathematics book series (SMM)

This monograph on the homotopy theory of topologized diagrams of spaces and spectra gives an expert account of a subject at the foundation of motivic homotopy theory and the theory of topological modular forms in stable homotopy theory.

Beginning with an introduction to the homotopy theory of simplicial sets and topos theory, the book covers core topics such as the unstable homotopy theory of simplicial presheaves and sheaves, localized theories, cocycles, descent theory, non-abelian cohomology, stacks, and local stable homotopy theory. A detailed treatment of the formalism of the subject is interwoven with explanations of the motivation, development, and nuances of ideas and results. The coherence of the abstract theory is elucidated through the use of widely applicable tools, such as Barr's theorem on Boolean localization, model structures on the category of simplicial presheaves on a site, and cocycle categories. A wealth of concrete examples convey the vitality and importance of the subject in topology, number theory, algebraic geometry, and algebraic *K*-theory.

Assuming basic knowledge of algebraic geometry and homotopy theory, *Local Homotopy Theory* will appeal to researchers and advanced graduate students seeking to understand and advance the applications of homotopy theory in multiple areas of mathematics and the mathematical sciences.

algebraic K-theory higher category theory homotopical algebra homotopy theory model categories motivic homotopy theory non-abelian cohomology simplicial sheaves stable homotopy theory topological modular forms

- DOI https://doi.org/10.1007/978-1-4939-2300-7
- Copyright Information Springer-Verlag New York 2015
- Publisher Name Springer, New York, NY
- eBook Packages Mathematics and Statistics
- Print ISBN 978-1-4939-2299-4
- Online ISBN 978-1-4939-2300-7
- Series Print ISSN 1439-7382
- Series Online ISSN 2196-9922
- About this book