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:
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 the coproduct $pt \sqcup pt$.
The subobject classifier $\Omega$ and the Sierpinski space $\mathbb{S}$ in the category of choice sets are bi-pointed objects.
In general, any non-degenerate subobject classifier in a topos or pretopos is a bi-pointed object.
Any interval object is a bi-pointed object with a 2-morphism connecting the two global elements.
The boolean domain is the initial bi-pointed object in Set.
From the bicategory structure on co-spans in $V$ bi-pointed objects in $V$ naturally inherit the structure of a monoidal category
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
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
where the morphism $pt \sqcup pt \stackrel{\sigma_X \sqcup \sigma_X}{\to} [X,Y]$ is adjunct to $X \otimes (pt \sqcup pt) \to pt \otimes (pt \sqcup pt) \simeq pt \sqcup pt \stackrel{\sigma_Y \sqcup \tau_Y}{\to} Y$.
Last revised on December 23, 2023 at 17:29:38. See the history of this page for a list of all contributions to it.