# Category Archives: Foundations

## 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

## A Formalized Interpreter

I’d like to announce a result that might be interesting: an interpreter for a very small dependent type system in Coq, assuming uniqueness of identity proofs (UIP). Because it assumes UIP, it’s not immediately compatible with HoTT, but it seems … Continue reading

## Higher inductive-recursive univalence and type-directed definitions

In chapter 2 of the HoTT book, we prove theorems (or, in a couple of cases, assert axioms) characterizing the equality types of all the standard type formers. For instance, we have and (that’s function extensionality) and (that’s univalence). However, … Continue reading

## Homotopy Type Theory should eat itself (but so far, it’s too big to swallow)

The title of this post is an homage to a well-known paper by James Chapman called Type theory should eat itself. I also considered titling the post How I spent my Christmas vacation trying to construct semisimplicial types in a … Continue reading

## Composition is not what you think it is! Why “nearly invertible” isn’t.

A few months ago, Nicolai Kraus posted an interesting and surprising result: the truncation map : ℕ → ‖ℕ‖ is nearly invertible. This post attempts to explain why “nearly invertible” is a misnomer. I, like many others, was very … Continue reading

## The HoTT Book

This posting is the official announcement of The HoTT Book, or more formally: Homotopy Type Theory: Univalent Foundations of Mathematics The Univalent Foundations Program, Institute for Advanced Study The book is the result of an amazing collaboration between virtually everyone involved … Continue reading

## Universe n is not an n-Type

Joint work with Christian Sattler Some time ago, at the UF Program in Princeton, I presented a proof that Universe n is not an n-type. We have now formalized that proof in Agda and want to present it here. One … Continue reading