The $0$-sphere1 is the pointed set $\smash {\webleft (S^{0},0\webright )}$2 consisting of:
- The Underlying Set. The set $S^{0}$ defined by
\[ S^{0} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ 0,1\webright\} . \]
- The Basepoint. The element $0$ of $S^{0}$.
1Further Terminology: In the context of monoids with zero as models for $\mathbb {F}_{1}$-algebras, the $0$-sphere is viewed as the underlying pointed set of the field with one element.
2Further Notation: In the context of monoids with zero as models for $\mathbb {F}_{1}$-algebras, $S^{0}$ is also denoted $\webleft (\mathbb {F}_{1},0\webright )$.
