The set with deleted basepoint associated to $X$ is the set $\smash {X^{-}}$ defined by
\[ X^{-}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X\setminus \left\{ x_{0}\right\} . \]
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1.
Functoriality. The assignment $\webleft (X,x_{0}\webright )\mapsto X^{-}$ defines a functor
\[ X^{-}\colon \mathsf{Sets}^{\mathrm{actv}}_{*}\to \mathsf{Sets}, \]where:
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Action on Objects. For each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}^{\mathrm{actv}}_{*}\webright )$, we have
\[ \webleft [\webleft (-\webright )^{-}\webright ]\webleft (X\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X^{-}, \]where $X^{-}$ is the set of Definition 6.4.2.1.1.
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Action on Morphisms. For each morphism $f\colon X\to Y$ of $\mathsf{Sets}^{\mathrm{actv}}_{*}$, the image
\[ f^{-}\colon X^{-}\to Y^{-} \]of $f$ by $\webleft (-\webright )^{-}$ is the map defined by
\[ f^{-}\webleft (x\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (x\webright ) \]for each $x\in X^{-}$.
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2.
Adjoint Equivalence. We have an adjoint equivalence of categories
witnessed by a bijection of sets
\begin{align*} \mathsf{Sets}\webleft (X^{-},Y\webright )\cong \mathsf{Sets}_{*}\webleft (X,Y^{+}\webright ),\end{align*}natural in $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$ and $Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$, and by isomorphisms
\begin{align*} \webleft (X^{-}\webright )^{+} & \cong X,\\ \webleft (Y^{+}\webright )^{-} & \cong Y, \end{align*}once again natural in $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$ and $Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$.
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3.
Symmetric Strong Monoidality With Respect to Wedge Sums. The functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\webleft (-\webright )^{-},\webleft (-\webright )^{-,\vee },\webleft (-\webright )^{-,\vee }_{\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets}^{\mathrm{actv}}_{*},\vee ,\mathrm{pt}\webright ), \to \webleft (\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\text{Ø}\webright ), \]being equipped with isomorphisms of pointed sets
\[ \begin{gathered} \webleft (-\webright )^{-,\vee }_{X,Y} \colon X^{-}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y^{-} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (X\vee Y\webright )^{-},\\ \webleft (-\webright )^{-,\vee }_{\mathbb {1}} \colon \text{Ø}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathrm{pt}^{-}, \end{gathered} \]natural in $X,Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$.
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4.
Symmetric Strong Monoidality With Respect to Smash Products. The free pointed set functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\webleft (-\webright )^{-},\webleft (-\webright )^{-,\times },\webleft (-\webright )^{-,\times }_{\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets}^{\mathrm{actv}}_{*},\wedge ,S^{0}\webright ), \to \webleft (\mathsf{Sets},\times ,\mathrm{pt}\webright ) \]being equipped with isomorphisms of pointed sets
\[ \begin{gathered} \webleft (-\webright )^{-}_{X,Y} \colon X^{-}\times Y^{-} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (X\wedge Y\webright )^{-},\\ \webleft (-\webright )^{-}_{\mathbb {1}} \colon \mathrm{pt}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (S^{0}\webright )^{-}, \end{gathered} \]natural in $X,Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$.
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Preservation of Identities. Let $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$. We have
\[ \operatorname {\mathrm{id}}^{-}_{X}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x \]for each $x\in X^{-}$, so $\operatorname {\mathrm{id}}^{-}_{X}=\operatorname {\mathrm{id}}_{X^{-}}$.
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Preservation of Composition. Given morphisms of pointed sets
\begin{align*} f & \colon \webleft (X,x_{0}\webright ) \to \webleft (Y,y_{0}\webright ),\\ g & \colon \webleft (Y,y_{0}\webright ) \to \webleft (Z,z_{0}\webright ), \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 $x\in X$, so $\webleft (g\circ f\webright )^{-}=g^{-}\circ f^{-}$.
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1.
Map I. We define a map
\[ \Phi _{X,Y}\colon \mathsf{Sets}\webleft (X^{-},Y\webright )\to \mathsf{Sets}^{\mathrm{actv}}_{*}\webleft (X,Y^{+}\webright ) \]by sending a map $\xi \colon X^{-}\to Y$ to the active morphism of pointed sets
\[ \xi ^{\dagger }\colon X\to Y^{+} \]given by
\[ \xi ^{\dagger }\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \xi \webleft (x\webright ) & \text{if $x\in X^{-}$,}\\ \star _{Y} & \text{if $x=x_{0}$,} \end{cases} \]for each $x\in X$, where this morphism is indeed active since $\xi \webleft (x\webright )\in Y=Y^{+}\setminus \left\{ \star _{Y}\right\} $ for all $x\in X^{-}$.
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2.
Map II. We define a map
\[ \Psi _{X,Y}\colon \mathsf{Sets}^{\mathrm{actv}}_{*}\webleft (X,Y^{+}\webright )\to \mathsf{Sets}\webleft (X^{-},Y\webright ) \]given by sending an active morphism of pointed sets $\xi \colon X\to Y^{+}$ to the map
\[ \xi ^{\dagger }\colon X^{-}\to Y \]defined by
\[ \xi ^{\dagger }\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\xi \webleft (x\webright ) \]for each $x\in X^{-}$, which is indeed well-defined (in that $\xi \webleft (x\webright )\in Y$ for all $x\in X^{-}$) since $\xi $ is active.
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3.
Invertibility I. Given a map of sets $\xi \colon X^{-}\to Y$, 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 )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Psi _{X,Y}\webleft (\left[\mspace {-6mu}\left[x\mapsto {\begin{cases} \xi \webleft (x\webright )& \text{if $x\in X^{-}$}\\ \star _{Y}& \text{if $x=x_{0}$}\end{cases}}\right]\mspace {-6mu}\right]\webright )\\ & = [\mspace {-3mu}[x\mapsto \xi \webleft (x\webright )]\mspace {-3mu}]\\ & = \xi \\ & = \webleft [\operatorname {\mathrm{id}}_{\mathsf{Sets}\webleft (X^{-},Y\webright )}\webright ]\webleft (\xi \webright ). \end{align*}Therefore we have
\[ \Psi _{X,Y}\circ \Phi _{X,Y}=\operatorname {\mathrm{id}}_{\mathsf{Sets}\webleft (X^{-},Y\webright )}. \] -
4.
Invertibility II. Given a morphism of pointed sets
\[ \xi \colon \webleft (X,x_{0}\webright )\to \webleft (Y^{+},\star _{Y}\webright ), \]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 ([\mspace {-3mu}[x\mapsto \xi \webleft (x\webright )]\mspace {-3mu}]\webright )\\ & = \left[\mspace {-6mu}\left[x\mapsto {\begin{cases} \xi \webleft (x\webright )& \text{if $x\in X^{-}$}\\ \star _{Y}& \text{if $x=x_{0}$}\end{cases}}\right]\mspace {-6mu}\right]\\ & = \xi \\ & = \webleft [\operatorname {\mathrm{id}}_{\mathsf{Sets}^{\mathrm{actv}}_{*}\webleft (X,Y^{+}\webright )}\webright ]\webleft (\xi \webright ). \end{align*}Therefore we have
\[ \Phi _{X,Y}\circ \Psi _{X,Y}=\operatorname {\mathrm{id}}_{\mathsf{Sets}^{\mathrm{actv}}_{*}\webleft (X,Y^{+}\webright )}. \] -
5.
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 map of sets $\xi \colon X'\to Y$, 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 )\\ & = \left[\mspace {-6mu}\left[x\mapsto {\begin{cases} \xi \webleft (f\webleft (x\webright )\webright )& \text{if $f\webleft (x\webright )\in X^{\prime ,-}$}\\ \star _{Y}& \text{if $f\webleft (x\webright )=x'_{0}$}\end{cases}}\right]\mspace {-6mu}\right]\\ & = f^{*}\webleft (\left[\mspace {-6mu}\left[x'\mapsto {\begin{cases} \xi \webleft (x'\webright )& \text{if $x'\in X^{\prime ,-}$}\\ \star _{Y}& \text{if $x'=x'_{0}$}\end{cases}}\right]\mspace {-6mu}\right]\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.
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6.
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 map of sets $\xi \colon X^{-}\to Y$, 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 )\\ & = \left[\mspace {-6mu}\left[x\mapsto {\begin{cases} g\webleft (\xi \webleft (x\webright )\webright )& \text{if $x\in X^{-}$}\\ \star _{Y'}& \text{if $x=x_{0}$}\end{cases}}\right]\mspace {-6mu}\right]\\ & = g_{*}\webleft (\left[\mspace {-6mu}\left[x\mapsto {\begin{cases} \xi \webleft (x\webright )& \text{if $x\in X^{-}$}\\ \star _{Y}& \text{if $x=x_{0}$}\end{cases}}\right]\mspace {-6mu}\right]\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.
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7.
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.
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8.
Fully Faithfulness of $\webleft (-\webright )^{-}$. We aim to show that the assignment $f\mapsto f^{-}$ sets up a bijection
\[ \webleft (-\webright )^{-}_{X,Y}\colon \mathsf{Sets}^{\mathrm{actv}}_{*}\webleft (X,Y\webright )\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathsf{Sets}\webleft (X^{-},Y^{-}\webright ). \]Indeed, the inverse map
\[ \webleft (-\webright )^{-,-1}_{X,Y}\colon \mathsf{Sets}\webleft (X^{-},Y^{-}\webright )\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathsf{Sets}^{\mathrm{actv}}_{*}\webleft (X,Y\webright ) \]is given by sending a map of sets $f\colon X^{-}\to Y^{-}$ to the active morphism of pointed sets $f^{\dagger }\colon X\to Y$ defined by
\[ f^{\dagger }\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} f\webleft (x\webright ) & \text{if $x\in X^{-}$,}\\ y_{0} & \text{if $x=x_{0}$} \end{cases} \]for each $x\in X$.
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9.
Essential Surjectivity of $\webleft (-\webright )^{-}$. We need to show that, given an object $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$, there exists some $X'\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}^{\mathrm{actv}}_{*}\webright )$ such that $\webleft (X'\webright )^{-}\cong X$. Indeed, taking $X'=X^{+}$, we have
\begin{align*} \webleft (X^{+}\webright )^{-} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (X\cup \left\{ \star _{X}\right\} \webright )^{-}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (X\cup \left\{ \star _{X}\right\} \webright )\setminus \left\{ \star _{X}\right\} \\ & = X, \end{align*}and thus we have in fact an equality $\webleft (X^{+}\webright )^{-}=X$, showing $\webleft (-\webright )^{-}$ to be essentially surjective.
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10.
The Functor $\webleft (-\webright )^{-}$ Is an Equivalence. Since $\webleft (-\webright )^{-}$ is fully faithful and essentially surjective, it is an equivalence by Chapter 11: Categories, Item 1 of Proposition 11.6.7.1.2.
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The Strong Monoidality Constraints. The isomorphism
\[ \webleft (-\webright )^{-,\vee }_{X,Y}\colon X^{-}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y^{-}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (X\vee Y\webright )^{-} \]is given by
\[ \webleft (-\webright )^{-,\vee }_{X,Y}\webleft (z\webright )=\begin{cases} \webleft [\webleft (0,x\webright )\webright ] & \text{if $z=\webleft (0,x\webright )$ with $x\in X$,}\\ \webleft [\webleft (1,y\webright )\webright ] & \text{if $z=\webleft (1,y\webright )$ with $y\in Y$}\end{cases} \]for each $z\in X^{-}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y^{-}$, with inverse
\[ \webleft (-\webright )^{-,\vee ,-1}_{X,Y} \colon \webleft (X\vee Y\webright )^{-} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }X^{-}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y^{-} \]given by
\[ \webleft (-\webright )^{-,\vee ,-1}_{X,Y}\webleft (z\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \webleft (0,x\webright ) & \text{if $z=\webleft [\webleft (0,x\webright )\webright ]$,}\\ \webleft (1,y\webright ) & \text{if $z=\webleft [\webleft (1,y\webright )\webright ]$,} \end{cases} \]for each $z\in \webleft (X\vee Y\webright )^{-}$.
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The Strong Monoidal Unity Constraint. The isomorphism
\[ \webleft (-\webright )^{+,\vee ,\mathbb {1}}_{X,Y}\colon \text{Ø}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathrm{pt}^{-} \]is an equality.
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The Strong Monoidality Constraints. The isomorphism
\[ \webleft (-\webright )^{-}_{X,Y}\colon X^{-}\times Y^{-}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (X\wedge Y\webright )^{-} \]is given by
\[ \webleft (-\webright )^{-}_{X,Y}\webleft (x,y\webright )=x\wedge y \]for each $\webleft (x,y\webright )\in X^{-}\times Y^{-}$, with inverse
\[ \webleft (-\webright )^{-,-1}_{X,Y} \colon \webleft (X\wedge Y\webright )^{-} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }X^{-}\times Y^{-} \]given by
\[ \webleft (-\webright )^{-,-1}_{X,Y}\webleft (x\wedge y\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (x,y\webright ) \]for each $x\wedge y\in \webleft (X\wedge Y\webright )^{-}$.
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The Strong Monoidal Unity Constraint. The isomorphism
\[ \webleft (-\webright )^{-,\mathbb {1}}_{X,Y}\colon \mathrm{pt}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (S^{0}\webright )^{-} \]is given by sending $\star $ to $1$.
6.4.2 Deleting Basepoints
Let $\webleft (X,x_{0}\webright )$ be a pointed set.
Let $\webleft (X,x_{0}\webright )$ be a pointed set.
Proof of Proposition 6.4.2.1.2.
Item 1: Functoriality
We claim that $\webleft (-\webright )^{-}$ is indeed a functor:
This finishes the proof.
Item 2: Adjoint Equivalence
We proceed in a few steps:
This finishes the proof.
Item 3: Symmetric Strong Monoidality With Respect to Wedge Sums
We construct the strong monoidal structure on $\webleft (-\webright )^{-}$ with respect to $\vee $ and $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$ as follows:
The verification that these isomorphisms satisfy the coherence conditions making the functor $\webleft (-\webright )^{-}$ into a symmetric strong monoidal functor is omitted.
Item 4: Symmetric Strong Monoidality With Respect to Smash Products
We construct the strong monoidal structure on $\webleft (-\webright )^{+}$ with respect to $\wedge $ and $\times $ as follows:
The verification that these isomorphisms satisfy the coherence conditions making the functor $\webleft (-\webright )^{+}$ into a symmetric strong monoidal functor is omitted.
