Category Archives: Foundations

The Cantor-Schröder-Bernstein Theorem for ∞-groupoids

The classical Cantor-Schröder-Bernstein Theorem (CSB) of set theory, formulated by Cantor and first proved by Bernstein, states that for any pair of sets, if there is an injection of each one into the other, then the two sets are in … Continue reading

Posted in Foundations | 8 Comments

Impredicative Encodings, Part 3

In this post I will argue that, improving on previous work of Awodey-Frey-Speight, (higher) inductive types can be defined using impredicative encodings with their full dependent induction principles — in particular, eliminating into all type families without any truncation hypotheses … Continue reading

Posted in Foundations, Higher Inductive Types | 50 Comments

Impredicative Encodings of Inductive Types in HoTT

I recently completed my master’s thesis under the supervision of Steve Awodey and Jonas Frey. A copy can be found here. Known impredicative encodings of various inductive types in System F, such as the type of natural numbers do not … Continue reading

Posted in Applications, Foundations, Uncategorized | 5 Comments

Parametricity, automorphisms of the universe, and excluded middle

Specific violations of parametricity, or existence of non-identity automorphisms of the universe, can be used to prove classical axioms. The former was previously featured on this blog, and the latter is part of a discussion on the HoTT mailing list. In a cooperation … Continue reading

Posted in Foundations | 7 Comments

Parametricity and excluded middle

Exercise 6.9 of the HoTT book tells us that, and assuming LEM, we can exhibit a function  such that is a non-identity function  I have proved the converse of this. Like in exercise 6.9, we assume univalence. Parametricity In a … Continue reading

Posted in Foundations | 13 Comments

Real-cohesive homotopy type theory

Two new papers have recently appeared online: Brouwer’s fixed-point theorem in real-cohesive homotopy type theory by me, and Adjoint logic with a 2-category of modes, by Dan Licata with a bit of help from me. Both of them have fairly … Continue reading

Posted in Applications, Foundations, Paper | 13 Comments

Universal Properties of Truncations

Some days ago at the HoTT/UF workshop in Warsaw (which was a great event!), I have talked about functions out of truncations. I have focussed on the propositional truncation , and I want to write this blog post in case … Continue reading

Posted in Foundations, Homotopy Theory, Models, Paper, Talk | 3 Comments