# Category Archives: Code

## Covering Spaces

Covering spaces are one of the important topics in classical homotopy theory, and this post summarizes what we have done in HoTT.  We have formulated the covering spaces and (re)proved the classification theorem based on (right) -sets, i.e., sets equipped with … Continue reading

Posted in Code, Higher Inductive Types, Homotopy Theory | 5 Comments

## Running Spheres in Agda, Part II

(Sorry for the long delay after the Part I of this post.) This post will summarize my work on defining spheres in arbitrary finite dimensions (Sⁿ) in Agda. I am going to use the tools for higher-order paths (discussed in … Continue reading

## On h-Propositional Reflection and Hedberg’s Theorem

Thorsten Altenkirch, Thierry Coquand, Martin Escardo, Nicolai Kraus Overview Hedberg’s theorem states that decidable equality of a type implies that the type is an h-set, meaning that it has unique equality proofs. We describe how the assumption can be weakened. … Continue reading

Posted in Code, Foundations | 17 Comments

## Isomorphism implies equality

Thierry Coquand and I have proved that, for a large class of algebraic structures, isomorphism implies equality (assuming univalence). A class of algebraic structures Structures in this class consist of a type, some operations on this type, and propositional axioms … Continue reading

Posted in Applications, Code, Univalence | 13 Comments

## Positive h-levels are closed under W

I was asked about my proof showing that positive h-levels are closed under W (assuming extensionality), so I decided to write a short note about it. W-types are defined inductively as follows (using Agda notation): data W (A : Set) … Continue reading

Posted in Code | 18 Comments

## Truncations and truncated higher inductive types

Truncation is a homotopy-theoretic construction that given a space and an integer n returns an n-truncated space together with a map in an universal way. More precisely, if i is the inclusion of n-truncated spaces into spaces, then n-truncation is … Continue reading

Posted in Code, Higher Inductive Types | 15 Comments

## Running Spheres in Agda, Part I

Running Spheres in Agda, Part I Introduction Where will we end up with if we generalize circles S¹ to spheres Sⁿ? Here is the first part of my journey to a 95% working definition for arbitrary finite dimension. The 5% … Continue reading