6.1.4 Elementary Properties of Pointed Sets

    Proposition 6.1.4.1.1 ▶ Elementary Properties of Pointed Sets

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

    1. 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. 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. 3.

      Failure To Be Cartesian Closed. The category $\mathsf{Sets}_{*}$ is not Cartesian closed.1

    4. 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 $\webleft (X,x_{0}\webright )\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$.

    5. 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:

  • (i)

    A pointed set $\webleft (X,x_{0}\webright )$ to $X$.

  • (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 $x\in f^{-1}\webleft (Y\setminus y_{0}\webright )$.

  • (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 $\webleft (X,\star \webright )$ 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

      \[ \xi ^{-1}_{f}\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright ) \]

      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 $x\in X$.

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

    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 Unresolved reference, Unresolved reference.

    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 Unresolved reference, Unresolved reference.

    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''?].

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