An Introduction to Univalent Foundations for Mathematicians

Univalent Foundations and the UniMath Library

We give a concise presentation of the Univalent Foundations of mathematics underlining the main ideas (section 1), followed by a discussion of the large-scale UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations, and the challenges one faces in designing such a library (section 2). This leads us to a general discussion about the links between architectur...

Homotopy type theory and Voevodsky's univalent foundations

Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. This has given rise to a new field, which has been christened “homotopy type theory”. In this direction, Vladimir Voevodsky observed that it is possible to model type theory using simplicial sets and that this model satisfies an additional property, cal...

Introduction - from type theory and homotopy theory to univalent foundations

Univalent Foundations Project

While working on the completion of the proof of the Bloch-Kato conjecture I have thought a lot about what to do next. Eventually I became convinced that the most interesting and important directions in current mathematics are the ones related to the transition into a new era which will be characterized by the widespread use of automated tools for proof construction and verification. I have star...

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...