Content created by Fredrik Bakke, Jonathan Prieto-Cubides, Egbert Rijke, Raymond Baker and maybemabeline.

Created on 2023-01-28.
Last modified on 2023-10-12.

module synthetic-homotopy-theory.spheres where
open import elementary-number-theory.natural-numbers

open import foundation.dependent-pair-types
open import foundation.function-types
open import foundation.identity-types
open import foundation.universe-levels

open import structured-types.pointed-types

open import synthetic-homotopy-theory.iterated-suspensions-of-pointed-types
open import synthetic-homotopy-theory.suspensions-of-types

open import univalent-combinatorics.standard-finite-types


The spheres are defined as iterated suspensions of the standard two-element type Fin 2.


sphere-Pointed-Type :   Pointed-Type lzero
sphere-Pointed-Type n = iterated-suspension-Pointed-Type n (Fin 2 , zero-Fin 1)

sphere :   UU lzero
sphere = type-Pointed-Type  sphere-Pointed-Type

north-sphere : (n : )  sphere n
north-sphere zero-ℕ = zero-Fin 1
north-sphere (succ-ℕ n) = north-suspension

south-sphere : (n : )  sphere n
south-sphere zero-ℕ = one-Fin 1
south-sphere (succ-ℕ n) = south-suspension

meridian-sphere :
  (n : )  sphere n  north-sphere (succ-ℕ n)  south-sphere (succ-ℕ n)
meridian-sphere n = meridian-suspension

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