  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.
