3.1.1 Foundations
A pointed set is equivalently:
- An $\mathbb {E}_{0}$-monoid in $\webleft (\mathrm{N}_{\bullet }\webleft (\mathsf{Sets}\webright ),\text{pt}\webright )$.
- A pointed object in $\webleft (\mathsf{Sets},\text{pt}\webright )$.
The $0$-sphere is the pointed set $\smash {\webleft (S^{0},0\webright )}$ consisting of:
The trivial pointed set is the pointed set $\webleft (\text{pt},\star \webright )$ consisting of:
- The Underlying Set. The punctual set $\text{pt}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \star \webright\} $.
- The Basepoint. The element $\star $ of $\text{pt}$.
The underlying pointed set of a semimodule $\webleft (M,\alpha _{M}\webright )$ is the pointed set $\webleft (M,0_{M}\webright )$.
The underlying pointed set of a module $\webleft (M,\alpha _{M}\webright )$ is the pointed set $\webleft (M,0_{M}\webright )$.