  Homotopy type theory is a new branch of mathematics, based on a recently
discovered connection between homotopy theory and type theory, which brings new
ideas into the very foundation of mathematics. On the one hand, Voevodsky's
subtle and beautiful "univalence axiom" implies that isomorphic structures can
be identified. On the other hand, "higher inductive types" provide direct,
logical descriptions of some of the basic spaces and constructions of homotopy
theory. Both are impossible to capture directly in classical set-theoretic
foundations, but when combined in homotopy type theory, they permit an entirely
new kind of "logic of homotopy types". This suggests a new conception of
foundations of mathematics, with intrinsic homotopical content, an "invariant"
conception of the objects of mathematics -- and convenient machine
implementations, which can serve as a practical aid to the working
mathematician. This book is intended as a first systematic exposition of the
basics of the resulting "Univalent Foundations" program, and a collection of
examples of this new style of reasoning -- but without requiring the reader to
know or learn any formal logic, or to use any computer proof assistant.
