Subsection 3.1.3 (009L): The Category of Pointed Sets

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The category of pointed sets is the category $\mathsf{Sets}_{*}$ defined equivalently as

The homotopy category of the $\infty $-category $\mathsf{Mon}_{\mathbb {E}_{0}}\webleft (\mathrm{N}_{\bullet }\webleft (\mathsf{Sets}\webright ),\text{pt}\webright )$ of ;
The category $\mathsf{Sets}_{*}$ of .

In detail, the category of pointed sets is the category $\mathsf{Sets}_{*}$ where

Objects. The objects of $\mathsf{Sets}_{*}$ are pointed sets;
Morphisms. The morphisms of $\mathsf{Sets}_{*}$ are morphisms of pointed sets;
Identities. For each $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, the unit map

\[ \mathbb {1}^{\mathsf{Sets}_{*}}_{\webleft (X,x_{0}\webright )} \colon \text{pt}\to \mathsf{Sets}_{*}\webleft (\webleft (X,x_{0}\webright ),\webleft (X,x_{0}\webright )\webright ) \]

of $\mathsf{Sets}_{*}$ at $\webleft (X,x_{0}\webright )$ is defined by ^[1]

\[ \text{id}^{\mathsf{Sets}_{*}}_{\webleft (X,x_{0}\webright )} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{X}; \]
Composition. For each $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, the composition map

\[ \circ ^{\mathsf{Sets}_{*}}_{\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )} \colon \mathsf{Sets}_{*}\webleft (\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\webright ) \times \mathsf{Sets}_{*}\webleft (\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\webright ) \to \mathsf{Sets}_{*}\webleft (\webleft (X,x_{0}\webright ),\webleft (Z,z_{0}\webright )\webright ) \]

of $\mathsf{Sets}_{*}$ at $\webleft (\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\webright )$ is defined by ^[2]

\[ g\mathbin {{\circ }^{\mathsf{Sets}_{*}}_{\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )}}f \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g\circ f. \]

Footnotes

[1] Note that $\text{id}_{X}$ is indeed a morphism of pointed sets, as we have $\text{id}_{X}\webleft (x_{0}\webright )=x_{0}$.

[2] Note that the composition of two morphisms of pointed sets is indeed a morphism of pointed sets, as we have

\begin{align*} g\webleft (f\webleft (x_{0}\webright )\webright ) & = g\webleft (y_{0}\webright )\\ & = z_{0}, \end{align*}

in terms of diagrams.

The Clowder Project

3.1.3 The Category of Pointed Sets

Footnotes