The free pointed set on $X$ is the pointed set $\smash {X^{+}}$ consisting of:
- The Underlying Set. The set $X^{+}$ defined by1
\begin{align*} X^{+} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\text{pt}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\webleft\{ \star \webright\} . \end{align*}
- The Basepoint. The element $\star $ of $X^{+}$.
1Further Notation: We sometimes write $\star _{X}$ for the basepoint of $X^{+}$ for clarity, specially when there are multiple free pointed sets involved in the current discussion.
