Subsection 6.1.4 (009P): Elementary Properties of Pointed Sets

Let $(X,x_{0})$ be a pointed set.

1.
Completeness. The category $\mathsf{Sets}_{*}$ of pointed sets and morphisms between them is complete, having in particular:
1. (a)
  Products, described as in Definition 6.2.3.1.1.
2. (b)
  Pullbacks, described as in Definition 6.2.4.1.1.
3. (c)
  Equalisers, described as in Definition 6.2.5.1.1.
2.
Cocompleteness. The category $\mathsf{Sets}_{*}$ of pointed sets and morphisms between them is cocomplete, having in particular:
1. (a)
  Coproducts, described as in Definition 6.3.3.1.1.
2. (b)
  Pushouts, described as in Definition 6.3.4.1.1;
3. (c)
  Coequalisers, described as in Definition 6.3.5.1.1.
3.
Failure To Be Cartesian Closed. The category $\mathsf{Sets}_{*}$ is not Cartesian closed.¹
4.
Morphisms From the Monoidal Unit. We have a bijection of sets²

\[ \mathsf{Sets}_{*}(S^{0},X) \cong X, \]

natural in $(X,x_{0})\in \operatorname {\mathrm{Obj}}(\mathsf{Sets}_{*})$, internalising also to an isomorphism of pointed sets

\[ \boldsymbol {\mathsf{Sets}}_{*}(S^{0},X) \cong (X,x_{0}), \]

again natural in $(X,x_{0})\in \operatorname {\mathrm{Obj}}(\mathsf{Sets}_{*})$.
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:
1. (a)
  From Pointed Sets to Sets With Partial Functions. The equivalence
  
  \[ \xi \colon \mathsf{Sets}_{*}\mathbin {\overset {\cong }{\rightarrow }}\mathsf{Sets}^{\mathrm{part.}} \]
  
  sends:
  1. (i)
    A pointed set $(X,x_{0})$ to $X$.
  2. (ii)
    A pointed function
    
    \[ f\colon (X,x_{0})\to (Y,y_{0}) \]
    
    to the partial function
    
    \[ \xi _{f}\colon X\to Y \]
    
    defined on $f^{-1}(Y\setminus y_{0})$ and given by
    
    \[ \xi _{f}(x)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f(x) \]
    
    for each $x\in f^{-1}(Y\setminus y_{0})$.
2. (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:
  1. (i)
    A set $X$ is to the pointed set $(X,\star )$ with $\star $ an element that is not in $X$.
  2. (ii)
    A partial function
    
    \[ f\colon X\to Y \]
    
    defined on $U\subset X$ to the pointed function
    
    \[ \xi ^{-1}_{f}\colon (X,x_{0})\to (Y,y_{0}) \]
    
    defined by
    
    \[ \xi _{f}(x)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} f(x) & \text{if $x\in U$,}\\ y_{0} & \text{otherwise.} \end{cases} \]
    
    for each $x\in X$.

¹The category $\mathsf{Sets}_{*}$ does admit a natural monoidal closed structure, however; see Chapter 7: Tensor Products of Pointed Sets.
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 $S^{0}$.
³Warning: This is not an isomorphism of categories, only an equivalence.

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 $(X,x_{0})$ 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

\[ \boldsymbol {\mathsf{Sets}}_{*}(S^{0},X) \cong (X,x_{0}) \]

follows by noting that $\Delta _{x_{0}}\colon S^{0}\to X$, the basepoint of $\boldsymbol {\mathsf{Sets}}_{*}(S^{0},X)$, 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''?].

The Clowder Project

6.1.4 Elementary Properties of Pointed Sets