nLab bi-pointed object

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 the coproduct $pt \sqcup pt$.

In more general monoidal categories

One can consider generalised elements out of an object $I$, where we do not require $I$ to be the terminal object, a bi-pointed object is a co-span from $I$ to itself, i.e. a diagram of this form:

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

In many cases, $I$ would be a tensor unit of a monoidal category, in which case the pointed objects are pointed objects in a monoidal category.

Examples

• 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.

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 $X \otimes (pt \sqcup pt) \to pt \otimes (pt \sqcup pt) \simeq pt \sqcup pt \stackrel{\sigma_Y \sqcup \tau_Y}{\to} Y$.

See also

• pointed object

• boolean domain object

• bi-pointed type

• bi-pointed set