nLab bi-pointed object



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

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

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

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


Closed structure

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

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

Assume that the terminal object ptpt is the tensor unit in VV.

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

pt[X,Y] pt ptpt σ Yτ Y [X,Y] σ X *×τ X * [ptpt,Y]. \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 ptptσ Yτ Y[ptpt,Y]pt \sqcup pt \stackrel{\sigma_Y \sqcup \tau_Y}{\to} [pt \sqcup pt,Y] is adjunct to pt(ptpt)ptptσ Yτ YY\pt \otimes (pt \sqcup pt) \to pt \sqcup pt \stackrel{\sigma_Y \sqcup \tau_Y}{\to} Y.

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

ptpt Id ptpt σ Xσ X σ Yτ Y [X,Y] σ X *×σ Y * [ptpt,Y], \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 ptptσ Xσ X[X,Y]pt \sqcup pt \stackrel{\sigma_X \sqcup \sigma_X}{\to} [X,Y] is adjunct to X(ptpt)pt(ptpt)ptptσ Yτ YY 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

Last revised on December 23, 2023 at 17:29:38. See the history of this page for a list of all contributions to it.