Steve Awodey holds the Dean’s Chair in Logic at Carnegie Mellon University, where he is Professor of Philosophy and Mathematics. A founder of Homotopy Type Theory, he co-organized a special research year on Univalent Foundations at the Institute for Advanced Study (Princeton). His numerous publications include the textbook Category Theory and the collaborative volume Homotopy Type Theory: Univalent Foundations of Mathematics. He serves on several journal editorial boards and is coordinating editor of the Journal of Symbolic Logic. He has held visiting appointments at the Poincaré Institute (Paris), Newton Institute (Cambridge), Hausdorff Institute (Bonn), and the Centre for Advanced Studies (Oslo), and is currently a Royal Society Wolfson Visiting Fellow at Cambridge University.

Chapter 1. Introduction.- Chapter 2. Cartesian cubical sets.- Chapter 3. The cofibration weak factorization system.- Chapter 4. The fibration weak factorization system.- Chapter 5. The weak equivalences.- Chapter 6. The Frobenius condition.- Chapter 7. A universal fibration.- Chapter 8. The equivalence extension property.- Chapter 9. The fibration extension property.