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.

中文翻译：

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整数作为更高的归纳类型

我们考虑在同伦类型理论（HoTT）中定义整数的问题。我们可以将整数的类型定义为有符号自然数（即使用余积），但其归纳原理使用起来非常不方便，因为它会导致案例爆炸。另一种方法是使用集合商，但在这里我们需要使用集合截断来避免非平凡的更高等式。这导致递归原则只允许我们将函数定义为集合（满足 UIP 的类型）。在本文中，我们使用小宇宙或双可逆映射来考虑更高的归纳类型。这些类型表示没有显式集截断的整数，相当于通常的余积表示。这是一个有趣的例子，因为它展示了如何在 HoTT 中处理一些一致性问题。我们讨论了由这项工作引发的一些开放性问题。证明已经使用立方 Agda 进行了正式验证。