Category Archives: Foundations
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
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
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
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
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
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
Universal properties without function extensionality
A universal property, in the sense of category theory, generally expresses that a map involving hom-sets, induced by composition with some canonical map(s), is an isomorphism. In type theory we express this using equivalences of hom-types. For instance, the universal … Continue reading
