3.1.4 Elementary Properties of Pointed Sets

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

  1. Completeness. The category $\mathsf{Sets}_{*}$ of pointed sets and morphisms between them is complete, having in particular:
    1. Products, described as in Definition 3.2.3.1.1;
    2. Pullbacks, described as in Definition 3.2.4.1.1;
    3. Equalisers, described as in Definition 3.2.5.1.1.
  2. Cocompleteness. The category $\mathsf{Sets}_{*}$ of pointed sets and morphisms between them is cocomplete, having in particular:
    1. Coproducts, described as in Definition 3.3.3.1.1;
    2. Pushouts, described as in Definition 3.3.4.1.1;
    3. Coequalisers, described as in Definition 3.3.5.1.1.
  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 sets[2]
    \[ \mathsf{Sets}_{*}\webleft (S^{0},X\webright ) \cong X, \]

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

    \[ \textbf{Sets}_{*}\webleft (S^{0},X\webright ) \cong \webleft (X,x_{0}\webright ), \]

    again natural in $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.

  5. Relation to Partial Functions. We have an equivalence of categories[3]
    \[ \mathsf{Sets}_{*}\mathrel {\smash {\overset {\scriptscriptstyle \text{eq.}}\cong }}\mathsf{Sets}^{\mathrm{part.}} \]

    between the category of pointed sets and pointed functions between them and the category of sets and partial functions between them, where:

    1. From Pointed Sets to Sets With Partial Functions. The equivalence
      \[ \xi \colon \mathsf{Sets}_{*}\mathbin {\overset {\cong }{\rightarrow }}\mathsf{Sets}^{\mathrm{part.}} \]

      sends:

      1. A pointed set $\webleft (X,x_{0}\webright )$ to $X$.
      2. 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 $x\in f^{-1}\webleft (Y\setminus y_{0}\webright )$.

    2. 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. A set $X$ is to the pointed set $\webleft (X,\star \webright )$ with $\star $ an element that is not in $X$.
      2. 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 $x\in X$.

Item 1: Completeness
This follows from (the proofs) of Definition 3.2.3.1.1, Definition 3.2.4.1.1, and Definition 3.2.5.1.1 and .
Item 2: Cocompleteness
This follows from (the proofs) of Definition 3.3.3.1.1, Definition 3.3.4.1.1, and Definition 3.3.5.1.1 and .
Item 3: Failure To Be Cartesian Closed
See [MSE 2855868].
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

\[ \textbf{Sets}_{*}\webleft (S^{0},X\webright ) \cong \webleft (X,x_{0}\webright ) \]

follows by noting that $\Delta _{x_{0}}\colon S^{0}\to X$, the basepoint of $\textbf{Sets}_{*}\webleft (S^{0},X\webright )$, 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 [MSE 884460].


Footnotes

[1] The category $\mathsf{Sets}_{*}$ does admit monoidal closed structures however; see .
[2] In 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 $S^{0}$.
[3] Dangerous Bend SymbolWarning: This is not an isomorphism of categories, only an equivalence.


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