The Clowder Project

• •

  Preservation of Identities. Let $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$. We have

  \[ \operatorname {\mathrm{id}}^{+}_{X}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} x & \text{if $x\in X$,}\\ \star _{X} & \text{if $x=\star _{X}$,} \end{cases} \]

  for each $x\in X^{+}$, so $\operatorname {\mathrm{id}}^{+}_{X}=\operatorname {\mathrm{id}}_{X^{+}}$.

• •

  Preservation of Composition. Given morphisms of sets

  \begin{align*} f & \colon X \to Y,\\ g & \colon Y \to Z, \end{align*}

  we have

  \begin{align*} \webleft [g^{+}\circ f^{+}\webright ]\webleft (x\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g^{+}\webleft (f^{+}\webleft (x\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g^{+}\webleft (f\webleft (x\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g\webleft (f\webleft (x\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [g\circ f\webright ]^{+}\webleft (x\webright )\end{align*}

  for each $x\in X$ and

  \begin{align*} \webleft [g^{+}\circ f^{+}\webright ]\webleft (\star _{X}\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g^{+}\webleft (f^{+}\webleft (\star _{X}\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g^{+}\webleft (\star _{Y}\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\star _{Z}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [g\circ f\webright ]^{+}\webleft (\star _{X}\webright ), \end{align*}

  so $\webleft (g\circ f\webright )^{+}=g^{+}\circ f^{+}$.

This finishes the proof.

• •

  Map I. We define a map

  \[ \Phi _{X,Y}\colon \mathsf{Sets}_{*}\webleft (X^{+},Y\webright )\to \mathsf{Sets}\webleft (X,Y\webright ) \]

  by sending a morphism of pointed sets

  \[ \xi \colon \webleft (X^{+},\star _{X}\webright )\to \webleft (Y,y_{0}\webright ) \]

  to the function

  \[ \xi ^{\dagger }\colon X\to Y \]

  given by

  \[ \xi ^{\dagger }\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\xi \webleft (x\webright ) \]

  for each $x\in X$.

• •

  Map II. We define a map

  \[ \Psi _{X,Y}\colon \mathsf{Sets}\webleft (X,Y\webright )\to \mathsf{Sets}_{*}\webleft (X^{+},Y\webright ) \]

  given by sending a function $\xi \colon X\to Y$ to the morphism of pointed sets

  \[ \xi ^{\dagger }\colon \webleft (X^{+},\star _{X}\webright )\to \webleft (Y,y_{0}\webright ) \]

  defined by

  \[ \xi ^{\dagger }\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \xi \webleft (x\webright ) & \text{if $x\in X$,}\\ y_{0} & \text{if $x=\star _{X}$} \end{cases} \]

  for each $x\in X^{+}$.

• •

  Invertibility I. Given a morphism of pointed sets

  \[ \xi \colon \webleft (X^{+},\star _{X}\webright )\to \webleft (Y,y_{0}\webright ), \]

  we have

  \begin{align*} \webleft [\Psi _{X,Y}\circ \Phi _{X,Y}\webright ]\webleft (\xi \webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Psi _{X,Y}\webleft (\Phi _{X,Y}\webleft (\xi \webright )\webright )\\ & = \Psi _{X,Y}\webleft (\xi ^{\dagger }\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left[\mspace {-6mu}\left[x\mapsto {\begin{cases} \xi ^{\dagger }\webleft (x\webright )& \text{if $x\in X$}\\ y_{0}& \text{if $x=\star _{X}$}\end{cases}}\right]\mspace {-6mu}\right]\\ & = \left[\mspace {-6mu}\left[x\mapsto {\begin{cases} \xi \webleft (x\webright )& \text{if $x\in X$}\\ y_{0}& \text{if $x=\star _{X}$}\end{cases}}\right]\mspace {-6mu}\right]\\ & = \xi \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\operatorname {\mathrm{id}}_{\mathsf{Sets}_{*}\webleft (X^{+},Y\webright )}\webright ]\webleft (\xi \webright ).\end{align*}

  Therefore we have

  \[ \Psi _{X,Y}\circ \Phi _{X,Y}=\operatorname {\mathrm{id}}_{\mathsf{Sets}_{*}\webleft (X^{+},Y\webright )}. \]
• •

  Invertibility II. Given a map of sets $\xi \colon X\to Y$, we have

  \begin{align*} \webleft [\Phi _{X,Y}\circ \Psi _{X,Y}\webright ]\webleft (\xi \webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{X,Y}\webleft (\Psi _{X,Y}\webleft (\xi \webright )\webright )\\ & = \Phi _{X,Y}\webleft (\xi ^{\dagger }\webright )\\ & = \Phi _{X,Y}\webleft (\left[\mspace {-6mu}\left[x\mapsto {\begin{cases} \xi \webleft (x\webright )& \text{if $x\in X$}\\ y_{0}& \text{if $x=\star _{X}$}\end{cases}}\right]\mspace {-6mu}\right]\webright )\\ & = [\mspace {-3mu}[x\mapsto \xi \webleft (x\webright )]\mspace {-3mu}]\\ & = \xi \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\operatorname {\mathrm{id}}_{\mathsf{Sets}\webleft (X,Y\webright )}\webright ]\webleft (\xi \webright ).\end{align*}

  Therefore we have

  \[ \Phi _{X,Y}\circ \Psi _{X,Y}=\operatorname {\mathrm{id}}_{\mathsf{Sets}\webleft (X,Y\webright )}. \]
• •

  Naturality for $\Phi $, Part I. We need to show that, given a morphism of pointed sets

  \[ f\colon \webleft (X,x_{0}\webright )\to \webleft (X',x'_{0}\webright ), \]

  the diagram

  
  commutes. Indeed, given a morphism of pointed sets $\xi \colon X^{\prime ,+}\to Y$, we have
  \begin{align*} \webleft [\Phi _{X,Y}\circ f^{*}\webright ]\webleft (\xi \webright ) & = \Phi _{X,Y}\webleft (f^{*}\webleft (\xi \webright )\webright )\\ & = \Phi _{X,Y}\webleft (\xi \circ f\webright )\\ & = \xi \circ f\\ & = \Phi _{X',Y}\webleft (\xi \webright )\circ f\\ & = f^{*}\webleft (\Phi _{X',Y}\webleft (\xi \webright )\webright )\\ & = f^{*}\webleft (\Phi _{X',Y}\webleft (\xi \webright )\webright )\\ & = \webleft [f^{*}\circ \Phi _{X',Y}\webright ]\webleft (\xi \webright ). \end{align*}

  Therefore we have

  \[ \Phi _{X,Y}\circ f^{*}=f^{*}\circ \Phi _{X',Y} \]

  and the naturality diagram for $\Phi $ above indeed commutes.

• •

  Naturality for $\Phi $, Part II. We need to show that, given a morphism of pointed sets

  \[ g\colon \webleft (Y,y_{0}\webright )\to \webleft (Y',y'_{0}\webright ), \]

  the diagram

  
  commutes. Indeed, given a morphism of pointed sets
  \[ \xi ^{\dagger }\colon X^{+} \to Y, \]

  we have

  \begin{align*} \webleft [\Phi _{X,Y'}\circ g_{*}\webright ]\webleft (\xi \webright ) & = \Phi _{X,Y'}\webleft (g_{*}\webleft (\xi \webright )\webright )\\ & = \Phi _{X,Y'}\webleft (g\circ \xi \webright )\\ & = g\circ \xi \\ & = g\circ \Phi _{X,Y'}\webleft (\xi \webright )\\ & = g_{*}\webleft (\Phi _{X,Y'}\webleft (\xi \webright )\webright )\\ & = \webleft [g_{*}\circ \Phi _{X,Y'}\webright ]\webleft (\xi \webright ). \end{align*}

  Therefore we have

  \[ \Phi _{X,Y'}\circ g_{*}=g_{*}\circ \Phi _{X,Y'} \]

  and the naturality diagram for $\Phi $ above indeed commutes.

• •

  Naturality for $\Psi $. Since $\Phi $ is natural in each argument and $\Phi $ is a componentwise inverse to $\Psi $ in each argument, it follows from Chapter 11: Categories, Item 2 of Proposition 11.9.7.1.2 that $\Psi $ is also natural in each argument.

This finishes the proof.

• •

  The Strong Monoidality Constraints. The isomorphism

  \[ \webleft (-\webright )^{+,\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{X,Y}\colon X^{+}\vee Y^{+}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )^{+} \]

  is given by

  \[ \webleft (-\webright )^{+,\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{X,Y}\webleft (z\webright )=\begin{cases} x & \text{if $z=\webleft [\webleft (0,x\webright )\webright ]$ with $x\in X$,}\\ y & \text{if $z=\webleft [\webleft (1,y\webright )\webright ]$ with $y\in Y$,}\\ \star _{X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y} & \text{if $z=\webleft [\webleft (0,\star _{X}\webright )\webright ]$,}\\ \star _{X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y} & \text{if $z=\webleft [\webleft (1,\star _{Y}\webright )\webright ]$}\end{cases} \]

  for each $z\in X^{+}\vee Y^{+}$, with inverse

  \[ \webleft (-\webright )^{+,\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},-1}_{X,Y} \colon \webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )^{+} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }X^{+}\vee Y^{+} \]

  given by

  \[ \webleft (-\webright )^{+,\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},-1}_{X,Y}\webleft (z\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \webleft [\webleft (0,x\webright )\webright ] & \text{if $z=\webleft [\webleft (0,x\webright )\webright ]$,}\\ \webleft [\webleft (1,y\webright )\webright ] & \text{if $z=\webleft [\webleft (1,y\webright )\webright ]$,}\\ p_{0} & \text{if $z=\star _{X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y}$} \end{cases} \]

  for each $z\in \webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )^{+}$.

• •

  The Strong Monoidal Unity Constraint. The isomorphism

  \[ \webleft (-\webright )^{+,\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\mathbb {1}}_{X,Y}\colon \mathrm{pt}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\text{Ø}^{+} \]

  is given by sending $\star _{X}$ to $\star _{\text{Ø}}$.

The verification that these isomorphisms satisfy the coherence conditions making the functor $\webleft (-\webright )^{+}$ into a symmetric strong monoidal functor is omitted.

• •

  The Strong Monoidality Constraints. The isomorphism

  \[ \webleft (-\webright )^{+}_{X,Y}\colon X^{+}\wedge Y^{+}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (X\times Y\webright )^{+} \]

  is given by

  \[ \webleft (-\webright )^{+}_{X,Y}\webleft (x\wedge y\webright )=\begin{cases} \webleft (x,y\webright ) & \text{if $x\neq \star _{X}$ and $y\neq \star _{Y}$}\\ \star _{X\times Y} & \text{otherwise}\end{cases} \]

  for each $x\wedge y\in X^{+}\wedge Y^{+}$, with inverse

  \[ \webleft (-\webright )^{+,-1}_{X,Y} \colon \webleft (X\times Y\webright )^{+} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }X^{+}\wedge Y^{+} \]

  given by

  \[ \webleft (-\webright )^{+,-1}_{X,Y}\webleft (z\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} x\wedge y & \text{if $z=\webleft (x,y\webright )$ with $\webleft (x,y\webright )\in X\times Y$,}\\ \star _{X}\wedge \star _{Y} & \text{if $z=\star _{X\times Y}$,} \end{cases} \]

  for each $z\in \webleft (X\times Y\webright )^{+}$.

• •

  The Strong Monoidal Unity Constraint. The isomorphism

  \[ \webleft (-\webright )^{+,\mathbb {1}}_{X,Y}\colon S^{0}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathrm{pt}^{+} \]

  is given by sending $0$ to $\star _{\mathrm{pt}}$ and $1$ to $\star $, where $\mathrm{pt}^{+}=\left\{ \star ,\star _{\mathrm{pt}}\right\} $.

The verification that these isomorphisms satisfy the coherence conditions making the functor $\webleft (-\webright )^{+}$ into a symmetric strong monoidal functor is omitted.