Subsection 4.1.1 (009A): Foundations

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4.1.1 Foundations

A pointed set¹ is equivalently:

An $\mathbb {E}_{0}$-monoid in $\webleft (\mathrm{N}_{\bullet }\webleft (\mathsf{Sets}\webright ),\text{pt}\webright )$.
A pointed object in $\webleft (\mathsf{Sets},\text{pt}\webright )$.

¹Further Terminology: In the context of monoids with zero as models for $\mathbb {F}_{1}$-algebras, pointed sets are viewed as $\mathbb {F}_{1}$-modules.

Remark 4.1.1.1.2 ▶ Unwinding Definition 4.1.1.1.1

In detail, a pointed set is a pair $\webleft (X,x_{0}\webright )$ consisting of:

The Underlying Set. A set $X$, called the underlying set of $\webleft (X,x_{0}\webright )$.
The Basepoint. A morphism

\[ \webleft [x_{0}\webright ]\colon \text{pt}\to X \]

in $\mathsf{Sets}$, determining an element $x_{0}\in X$, called the basepoint of $X$.

Example 4.1.1.1.3 ▶ The Zero Sphere

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 $\webleft (\mathbb {F}_{1},0\webright )$.

Example 4.1.1.1.4 ▶ The Trivial Pointed Set

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 $\text{pt}$.

Example 4.1.1.1.5 ▶ The Standard Pointed Set With $n+1$ Elements

The standard pointed set with $n+1$ elements is the pointed set $\left\langle n\right\rangle $ consisting of

The Underlying Set. The set $\left\langle n\right\rangle $ defined by

\[ \left\langle n\right\rangle \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ *\webright\} \cup \webleft\{ 1,\ldots ,n\webright\} . \]
The Basepoint. The element $*$ of $\left\langle n\right\rangle $.