The Clowder Project

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

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

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 $(X,x_{0})$ 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 $x\in X$.