Abstract

We consider the problem of defining the integers in Homotopy Type Theory (HoTT). We can define the type of integers as signed natural numbers (i.e., using a coproduct), but its induction principle is very inconvenient to work with, since it leads to an explosion of cases. An alternative is to use set-quotients, but here we need to use set-truncation to avoid non-trivial higher equalities. This results in a recursion principle that only allows us to define function into sets (types satisfying UIP). In this paper we consider higher inductive types using either a small universe or bi-invertible maps. These types represent integers without explicit set-truncation that are equivalent to the usual coproduct representation. This is an interesting example since it shows how some coherence problems can be handled in HoTT. We discuss some open questions triggered by this work. The proofs have been formally verified using cubical Agda.

Highlights

  • How to define the integers in Homotopy Type Theory (HoTT)? This can sound like a trivial question

  • We use set-quotients, which can be implemented as a higher inductive type with a set-truncation constructor [18, Section 6.10]

  • The set-truncation constructor implies that using its recursion principle we can only define functions into sets, which seems to be an unreasonable limitation when working in HoTT

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Summary

Introduction

How to define the integers in HoTT? This can sound like a trivial question. The first answer is as signed natural numbers: Definition 1.1. Let Zb be the higher inductive type with the following constructors: For this definition we can give a complete proof that Zb is equivalent to Zw , which has been formalized in cubical Agda. The problem of defining the integers with convenient constructors, and adding only the right coherences to make it a set, can be seen as a simple instance of a more general class of coherence problems in HoTT. Another example that we have in mind is the intrinsic definition of the syntax of type theory as the initial category with families as developed in [3]. We hope that in this case we can add the correct coherence laws and show that they are sufficient to deduce that the initial algebra is a set

Contributions
Related work
Background
Representing Z using bi-invertible maps
Z is a set
Representing Z using a universe
Formalization in cubical Agda
Open questions
Full Text
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