Injective types in univalent mathematics

Univalent Foundations of Mathematics

Homotopy Type Theory: Univalent Foundations of Mathematics

These lecture notes are based on and partly contain material from the HoTT book and are licensed under Creative Commons Attribution-ShareAlike 3.0.

Homotopy Type Theory: Univalent Foundations of Mathematics

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 descriptio...

Types in Logic and Mathematics before 1940

In this article, we study the prehistory of type theory up to 1910 and its development betweenRussell andWhitehead’sPrincipiaMathematica ([71], 1910–1912) andChurch’s simply typed ë-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by desc...

Types in logic and mathematics before 1940

In this article, we study the prehistory of type theory up to 1910 and its development betweenRussell andWhitehead’sPrincipiaMathematica ([71], 1910–1912) andChurch’s simply typed ë-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by desc...