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\begin{document}

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\section*{pointed set}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{the_category_of_pointed_sets}{The category of pointed sets}\dotfill \pageref*{the_category_of_pointed_sets} \linebreak
\noindent\hyperlink{definition_2}{Definition}\dotfill \pageref*{definition_2} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{interpretation_as_universal_setbundle}{Interpretation as universal Set-bundle}\dotfill \pageref*{interpretation_as_universal_setbundle} \linebreak
\noindent\hyperlink{interpretation_as_2subobjectclassfier}{Interpretation as 2-subobject-classfier}\dotfill \pageref*{interpretation_as_2subobjectclassfier} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{pointed set} is a [[pointed object]] in [[Set]], hence a [[set]] $S$ equipped with a chosen element $s$ of $S$. (Compare [[inhabited set]], where the element is not specified.)

Since we can identify a (set-theoretic) element of $S$ with a (category-theoretic) [[global element]] (a morphism $s: 1 \to S$), we see that a pointed set is an object of the [[under category]] $\pt \downarrow \Set$, or coslice category $1/\Set$, of objects under the [[singleton]] $\{\bullet\}$.

\hypertarget{the_category_of_pointed_sets}{}\subsection*{{The category of pointed sets}}\label{the_category_of_pointed_sets}

\hypertarget{definition_2}{}\subsubsection*{{Definition}}\label{definition_2}

\begin{udefn}
The [[category]] $Set_*$ of pointed sets is the [[under category]] $*/Set$ of [[Set]] under the singleton set.

\end{udefn}
So a morphism $(S_1, s_1) \to (S_2, s_2)$ is a map between sets which maps these chosen elements to each other, i.e., commuting triangles

\begin{displaymath}
\itexarray{
    && pt
    \\
    & \swarrow && \searrow
    \\
    S_1
    &&\to&&
    S_2
  }
 \,.
\end{displaymath}
The category $Set_*$ naturally comes with [[forgetful functor]] $p : Set_* \to Set$ which forgets the tip of these triangles.

\hypertarget{properties}{}\subsubsection*{{Properties}}\label{properties}

\begin{uprop}
Equipped with the [[smash product]] $\otimes := \wedge$ of pointed set, $(Set_*, \wedge)$ is a [[closed monoidal category|closed]] [[symmetric monoidal category]].

The [[internal hom]] $Set_*(X,Y)$ is the [[hom-set]] in $*/Set$ pointed by the morphism $X \to Y$ that sends everything to the basepoint in $Y$.

\end{uprop}
See at \emph{[[pointed object]]} for more details.

\hypertarget{interpretation_as_universal_setbundle}{}\paragraph*{{Interpretation as universal Set-bundle}}\label{interpretation_as_universal_setbundle}

The morphism $Set_* \to Set$ is an example of a [[generalized universal bundle]]: the \emph{universal [[Set]]-bundle}. The entire structure here can be understood as arising from the (strict) pullback diagram

\begin{displaymath}
\itexarray{
    Set_* &\to& pt
    \\
    \downarrow && \downarrow^{pt \mapsto pt}
    \\
    [I,Set]
    &\stackrel{d_0}{\to}& Set
    \\
    \downarrow^{d_1}
    \\
    Set
  }
\end{displaymath}
in the 1-category [[Cat]], where

\begin{itemize}%
\item $I = \{a \to b\}$ is the [[interval category]];


\item $[I, Set] = Arr(Set)$ is the [[closed monoidal category|internal hom]] category which here is the [[arrow category]] of $Set$;


\item $d_i := [j_i, Set]$ are the images of the two injections $j_i : pt \to I$ of the point to the left and the right end of the interval, respectively -- so these functors evaluate on the left and right end of the interval, respectively;


\item the square is a pullback;


\item the total vertical functor is the forgetful functor $p : Set_* \to Set$.



\end{itemize}
The way in which $Set_* \to Set$ is the ``universal Set-bundle'' is discussed pretty explicitly in

\begin{itemize}%
\item Kathryn Hess, \emph{[[HessLackBundCat.pdf:file]]} .

\end{itemize}
(The discussion there becomes more manifestly one of bundles if one regards all morphisms $C \to Set$ appearing there as being the right legs of [[anafunctor]]s. )

\hypertarget{interpretation_as_2subobjectclassfier}{}\paragraph*{{Interpretation as 2-subobject-classfier}}\label{interpretation_as_2subobjectclassfier}

Observing that usual morphism into the [[subobject classifier]] $\Omega$ of the [[topos]] [[Set]] is the [[universal truth-value bundle]] $\{\top\} \to \TV$, and noticing that $TV = (-1)Cat$ and $Set = 0Cat$ suggests that $Set_* \to Set$ is a [[vertical categorification|categorified]] subobject classifier: indeed, it is the subobject classifier in the [[2-topos]] [[Cat]].

For discussion of this point see

\begin{itemize}%
\item David Corfield: \emph{101 things to do with a 2-classifier} (\href{http://golem.ph.utexas.edu/category/2008/01/101_things_to_do_with_a_2class.html}{blog})

\end{itemize}
It was David Roberts who pointed out in

\begin{itemize}%
\item David Roberts, \href{http://golem.ph.utexas.edu/category/2008/01/101_things_to_do_with_a_2class.html#c014559}{comment to: 101 things to do with a 2-classifier}

\end{itemize}
the relation between these higher classifiers and higher [[generalized universal bundle]]s, motivated by the observations on principal universal 1- and 2-bundles in

\begin{itemize}%
\item David Roberts, Urs Schreiber, \emph{The inner automorphism 3-group of a strict 2-group}, Journal of Homotopy and Related Structures, Vol. 3(2008), No. 1, pp. 193-244, (\href{http://arxiv.org/abs/0708.1741v2}{arXiv}).

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[pointed object]]


\item [[pointed topological space]]


\item [[pointed simplicial set]]


\item [[pointed homotopy type]]



\end{itemize}
[[!redirects pointed sets]]

[[!redirects category of pointed sets]]



\end{document}