## Idea Spheres can be thought of as a space generated by a point and a (higher-)[[path]] constructor. This does not mean their homotopical information is simple however. In face it is very much still an open problem to determine the [[homotopy groups of spheres]]. ## Definition The $n$-sphere can be defined as the [[suspension]] of the $(n-1)$-sphere or the iterated suspension of $S^0 := \mathbf{2}$. (a.k.a $\text{Bool}$). $$S^n := \Sigma^n S^0$$ ## Properties ## See also * [[homotopy groups of spheres]] * [[Synthetic homotopy theory]] ## References * [[HoTT book]]