nLab bi-pointed object


A bi-pointed object in a category VV with terminal object ptpt is a co-span from ptpt to itself, i.e. a diagram

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 \emptyset and terminal object ptpt is a co-span from \emptyset to ptpt, and if VV has in addition binary coproducts then a bi-pointed object in VV is the same as a co-span from \emptyset to 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} & \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 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 &\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 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 March 21, 2021 at 14:47:04. See the history of this page for a list of all contributions to it.