Let $\webleft (X,x_{0}\webright )$ be a pointed set.
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1.
Completeness. The category $\mathsf{Sets}_{*}$ of pointed sets and morphisms between them is complete, having in particular:
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(a)
Products, described as in Definition 6.2.3.1.1.
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(b)
Pullbacks, described as in Definition 6.2.4.1.1.
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(c)
Equalisers, described as in Definition 6.2.5.1.1.
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(a)
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2.
Cocompleteness. The category $\mathsf{Sets}_{*}$ of pointed sets and morphisms between them is cocomplete, having in particular:
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(a)
Coproducts, described as in Definition 6.3.3.1.1.
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(b)
Pushouts, described as in Definition 6.3.4.1.1;
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(c)
Coequalisers, described as in Definition 6.3.5.1.1.
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(a)
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3.
Failure To Be Cartesian Closed. The category $\mathsf{Sets}_{*}$ is not Cartesian closed.1
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4.
Morphisms From the Monoidal Unit. We have a bijection of sets2
\[ \mathsf{Sets}_{*}\webleft (S^{0},X\webright ) \cong X, \]natural in $\webleft (X,x_{0}\webright )\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$, internalising also to an isomorphism of pointed sets
\[ \boldsymbol {\mathsf{Sets}}_{*}\webleft (S^{0},X\webright ) \cong \webleft (X,x_{0}\webright ), \]again natural in $\webleft (X,x_{0}\webright )\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$.
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5.
Relation to Partial Functions. We have an equivalence of categories3
\[ \mathsf{Sets}_{*}\mathrel {\smash {\overset {\scriptscriptstyle \text{eq.}}\cong }}\mathsf{Sets}^{\mathrm{part.}} \]between the category of pointed sets and pointed functions between them and the category of sets and partial functions between them, where:
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(a)
From Pointed Sets to Sets With Partial Functions. The equivalence
\[ \xi \colon \mathsf{Sets}_{*}\mathbin {\overset {\cong }{\rightarrow }}\mathsf{Sets}^{\mathrm{part.}} \]sends:
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(i)
A pointed set $\webleft (X,x_{0}\webright )$ to $X$.
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(ii)
A pointed function
\[ f\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright ) \]to the partial function
\[ \xi _{f}\colon X\to Y \]defined on $f^{-1}\webleft (Y\setminus y_{0}\webright )$ and given by
\[ \xi _{f}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (x\webright ) \]for each $x\in f^{-1}\webleft (Y\setminus y_{0}\webright )$.
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(i)
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(b)
From Sets With Partial Functions to Pointed Sets. The equivalence
\[ \xi ^{-1}\colon \mathsf{Sets}^{\mathrm{part.}}\mathbin {\overset {\cong }{\rightarrow }}\mathsf{Sets}_{*} \]sends:
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(i)
A set $X$ is to the pointed set $\webleft (X,\star \webright )$ with $\star $ an element that is not in $X$.
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(ii)
A partial function
\[ f\colon X\to Y \]defined on $U\subset X$ to the pointed function
\[ \xi ^{-1}_{f}\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright ) \]defined by
\[ \xi _{f}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} f\webleft (x\webright ) & \text{if $x\in U$,}\\ y_{0} & \text{otherwise.} \end{cases} \]for each $x\in X$.
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(i)
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(a)
- 1The category $\mathsf{Sets}_{*}$ does admit a natural monoidal closed structure, however; see Chapter 7: Tensor Products of Pointed Sets .
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2In other words, the forgetful functor \[ {\text{忘}}\colon \mathsf{Sets}_{*}\to \mathsf{Sets} \]defined on objects by sending a pointed set to its underlying set is corepresentable by $S^{0}$.
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3
Warning: This is not an isomorphism of categories, only an equivalence.
6.1.4 Elementary Properties of Pointed Sets
Proof of Proposition 6.1.4.1.1.
Item 1: Completeness
This follows from (the proofs) of Definition 6.2.3.1.1, Definition 6.2.4.1.1, and Definition 6.2.5.1.1 and 
, .
Item 2: Cocompleteness
This follows from (the proofs) of Definition 6.3.3.1.1, Definition 6.3.4.1.1, and Definition 6.3.5.1.1 and , .
Item 3: Failure To Be Cartesian Closed
See [Yuan, Is the category of pointed sets Cartesian closed?].
Item 4: Morphisms From the Monoidal Unit
Since a morphism from $S^{0}$ to a pointed set $\webleft (X,x_{0}\webright )$ sends $0\in S^{0}$ to $x_{0}$ and then can send $1\in S^{0}$ to any element of $X$, we obtain a bijection between pointed maps $S^{0}\to X$ and the elements of $X$.
The isomorphism then
\[ \boldsymbol {\mathsf{Sets}}_{*}\webleft (S^{0},X\webright ) \cong \webleft (X,x_{0}\webright ) \]
follows by noting that $\Delta _{x_{0}}\colon S^{0}\to X$, the basepoint of $\boldsymbol {\mathsf{Sets}}_{*}\webleft (S^{0},X\webright )$, corresponds to the pointed map $S^{0}\to X$ picking the element $x_{0}$ of $X$, and thus we see that the bijection between pointed maps $S^{0}\to X$ and elements of $X$ is compatible with basepoints, lifting to an isomorphism of pointed sets.
Item 5: Relation to Partial Functions
See [Brandenburg, Why are the category of pointed sets and the category of sets and partial functions ``essentially the same''?].
