The extra identifications are one of the problems, but a more
fundamental problem is that while the north pole has been identified
with the south pole, no other pairs of antipodal points have been
identified.

Dan

On Aug 18, 2018, Guillaume Brunerie <guillaume.brunerie@gmail.com> wrote:

> As Dan said, this is not a correct definition of RP^n.
>
> More specifically, the issue is that you identified the points "map south" and "map north" in two different ways: "glue south north" and
> "inv (glue north south)".
> You could try to add another constructor
> glue² : (x y : S^n) -> glue x y = inv (glue y x)
> but of course you're now again identifying them in two different ways, so you've just shifted the problem one dimension up.
>
> Den lör 18 aug. 2018 16:46Dan Christensen <jdc@uwo.ca> skrev:
>
>  On Aug 18, 2018, Ali Caglayan <alizter@gmail.com> wrote:
>
>  > This screams to me an alternative definition for ℝℙⁿ (and maybe ℂℙⁿ,...)
>  >
>  > Inductive RP (n:ℕ) :=
>  > | map : Sⁿ -> RP n
>  > | glue : (x y:S⁰) -> map(in x) = map(in y)
>  >
>  > Where in : S⁰→Sⁿ is the obvious inclusion map.
>
>  That HIT would be S^n with four edges attached: two edges connecting
>  the north pole to the south pole, and a self-loop at each pole. So
>  homotopically it would be S^n wedged with a wedge of four circles.
>
>  To get the right space with the right bundle over it, see:
>
>  The real projective spaces in homotopy type theory
>  Ulrik Buchholtz, Egbert Rijke
>  https://arxiv.org/abs/1704.05770
>
>  Dan
>
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