The Clowder Project

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.

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

Morphisms From the Monoidal Unit. We have a bijection of sets²

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

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

Relation to Partial Functions. We have an equivalence of categories³

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

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

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 $\webleft (X,\star \webright )$ 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 $x\in X$.