category theory

# Contents

## Definition

In evident variation of the notion of pointed objects, a bi-pointed object in a category $V$ with terminal object $pt$ is a co-span from $pt$ to itself, i.e. a diagram of this form:

$\array{ && S \\ & {}^{\sigma_S}\nearrow && \nwarrow^{\tau_S} \\ pt &&&& pt } \,.$

Similarly, a pointed object in a category with initial object $\varnothing$ and terminal object $pt$ may be regarded as a co-span from $\varnothing$ to $pt$.

If $V$ has in addition binary coproducts then a bi-pointed object in $V$ is the same as a co-span from $\varnothing$ to to the coproduct $pt \sqcup pt$.

## Closed structure

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

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

Assume that the terminal object $pt$ 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 ${}_{pt}[X,Y]_{pt} \in Obj(V)$ of bipointed objects $X$ and $Y$ defined as the pullback

$\array{ {}_{pt}[X,Y]_{pt} & \longrightarrow & pt \sqcup pt \\ \Big\downarrow && \Big\downarrow\mathrlap{{}^{\sigma_Y \sqcup \tau_Y}} \\ [X,Y] & \underset {\sigma_X^* \times \tau_X^*} {\longrightarrow} & [pt \sqcup pt,Y].}$

Here the map $pt \sqcup pt \stackrel{\sigma_Y \sqcup \tau_Y}{\to} [pt \sqcup pt,Y]$ is adjunct to $\pt \otimes (pt \sqcup pt) \to pt \sqcup pt \stackrel{\sigma_Y \sqcup \tau_Y}{\to} Y$.

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

$\array{ pt \sqcup pt & \overset{Id}{\longrightarrow}& pt \sqcup pt \\ \Big\downarrow\mathrlap{{}^{\sigma_X \sqcup \sigma_X}} && \Big\downarrow\mathrlap{{}^{\sigma_Y \sqcup \tau_Y}} \\ [X,Y] & \underset {\sigma_X^* \times \sigma_Y^*} {\longrightarrow} & [pt \sqcup pt, Y], }$

where the morphism $pt \sqcup pt \stackrel{\sigma_X \sqcup \sigma_X}{\to} [X,Y]$ is adjunct to $X \otimes (pt \sqcup pt) \to pt \otimes (pt \sqcup pt) \simeq pt \sqcup pt \stackrel{\sigma_Y \sqcup \tau_Y}{\to} Y$.