nLab
bi-pointed object

Definition

A bi-pointed object in a category $V$ with terminal object $\mathrm{pt}$ is a co-span from $\mathrm{pt}$ to itself, i.e. a diagram

$$\begin{array}{ccc}& & S\\ & {}^{{\sigma }_S}\nearrow & & {\nwarrow }^{{\tau }_S}\\ \mathrm{pt}& & & & \mathrm{pt}\end{array}.$$\array{ && S \\ & {}^{\sigma_S}\nearrow && \nwarrow^{\tau_S} \\ pt &&&& pt } \,.

Similarly, a pointed object in a category with initial object $\varnothing$ and terminal object $\mathrm{pt}$ is a co-span from $\varnothing$ to $\mathrm{pt}$, and if $V$ has in addition binary coproducts then a bi-pointed object in $V$ is the same as a co-span from $\varnothing$ to $\mathrm{pt}\bigsqcup \mathrm{pt}$.

Examples

• An interval object is a bi-pointed object.

Closed structure

From the bicategory structure on co-spans in $V$ bi-pointed objects in $V$ naturally inherit the structure of a monoidal category

$$\mathrm{BiPointed}(V)={\mathrm{End}}_{\mathrm{CoSpan}(V)}(\mathrm{pt}).$$BiPointed(V) = End_{CoSpan(V)}(pt) \,.

Assume that the terminal object $\mathrm{pt}$ is the tensor unit in $V$.

Then moreover, following the construction of the $V$-internal hom of pointed objects and being a special case of that of co-spans in $V$, there is an internal hom-object ${}_{\mathrm{pt}}[X,Y{]}_{\mathrm{pt}}\in \mathrm{Obj}(V)$ of bipointed objects $X$ and $Y$ defined as the pullback

$$\begin{array}{ccc}{}_{\mathrm{pt}}[X,Y{]}_{\mathrm{pt}}& \to & \mathrm{pt}\bigsqcup \mathrm{pt}\\ \downarrow & & {\downarrow }^{{\sigma }_Y\bigsqcup {\tau }_Y}\\ [X,Y]& \stackrel{{\sigma }_X^{*}\times {\tau }_X^{*}}{\to }& [\mathrm{pt}\bigsqcup \mathrm{pt},Y]\end{array}.$$\array{ {}_{pt}[X,Y]_{pt} & \rightarrow & pt \sqcup pt\\ \downarrow && \downarrow^{\sigma_Y \sqcup \tau_Y} \\ [X,Y] & \stackrel{\sigma_X^* \times \tau_X^*} {\rightarrow} & [pt \sqcup pt,Y]} \,.

Here the map $\mathrm{pt}\bigsqcup \mathrm{pt}\stackrel{{\sigma }_Y\bigsqcup {\tau }_Y}{\to }[\mathrm{pt}\bigsqcup \mathrm{pt},Y]$ is adjunct to $pt\otimes (\mathrm{pt}\bigsqcup \mathrm{pt})\to \mathrm{pt}\bigsqcup \mathrm{pt}\stackrel{{\sigma }_Y\bigsqcup {\tau }_Y}{\to }Y$.

This $V$-object ${}_{\mathrm{pt}}[X,Y{]}_{\mathrm{pt}}$ is itself naturally bi-pointed with the bi-point $\mathrm{pt}\bigsqcup \mathrm{pt}\to {}_{\mathrm{pt}}[X,Y{]}_{\mathrm{pt}}$ given by the morphism induced from the above pullback diagram by the commuting diagram

$$\begin{array}{ccc}\mathrm{pt}\bigsqcup \mathrm{pt}& \stackrel{\mathrm{Id}}{\to }& \mathrm{pt}\bigsqcup \mathrm{pt}\\ {\downarrow }^{{\sigma }_X\bigsqcup {\sigma }_X}& & {\downarrow }^{{\sigma }_Y\bigsqcup {\tau }_Y}\\ [X,Y]& \stackrel{{\sigma }_X^{*}\times {\sigma }_Y^{*}}{\to }& [\mathrm{pt}\bigsqcup \mathrm{pt},Y]\end{array},$$\array{ pt \sqcup pt &\stackrel{Id}{\to}& pt \sqcup pt \\ \downarrow^{\sigma_X \sqcup \sigma_X} && \downarrow^{\sigma_Y \sqcup \tau_Y} \\ [X,Y] &\stackrel{\sigma_X^* \times \sigma_Y^*}{\to}& [pt \sqcup pt, Y] } \,,

where the morphism $\mathrm{pt}\bigsqcup \mathrm{pt}\stackrel{{\sigma }_X\bigsqcup {\sigma }_X}{\to }[X,Y]$ is adjunct to $X\otimes (\mathrm{pt}\bigsqcup \mathrm{pt})\to \mathrm{pt}\otimes (\mathrm{pt}\bigsqcup \mathrm{pt})\simeq \mathrm{pt}\bigsqcup \mathrm{pt}\stackrel{{\sigma }_Y\bigsqcup {\tau }_Y}{\to }Y$