A 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 $s: {\ast} \to S$ from the terminal object ${\ast}$), we see that a pointed set is an object of the under category $\pt \downarrow \Set$, or coslice category ${\ast}/\Set$, of objects under the singleton set ${\ast}$.
The category $\Set_{\ast}$ of pointed sets is the under category ${\ast}/\Set$ of Set under the singleton set ${\ast}$.
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
The category $\Set_{\ast}$ naturally comes with a forgetful functor $p : \Set_{\ast} \to \Set$ which forgets the tip of these triangles.
Equipped with the smash product ${\otimes} := {\wedge}$ of pointed sets, $(\Set_{\ast}, {\wedge})$ is a closed symmetric monoidal category.
The internal hom $\Set_{\ast}(X,Y)$ is the hom-set in ${\ast}/\Set$ pointed by the morphism $X \to Y$ that sends everything to the basepoint in $Y$.
See at pointed object for more details.
The morphism $\Set_{\ast} \to \Set$ is an example of a generalized universal bundle: the universal Set-bundle. The entire structure here can be understood as arising from the (strict) pullback diagram
in the 1-category Cat, where
$I = \{0 \to 1\}$ is the interval category;
$[I, \Set] = Arr(\Set)$ is the internal hom category which here is the arrow category of $\Set$;
$d_i := [j_i, \Set]$ 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;
the square is a pullback;
the total vertical functor is the forgetful functor $p : \Set_{\ast} \to \Set$.
The way in which $\Set_{\ast} \to \Set$ is the “universal Set-bundle” is discussed pretty explicitly in
(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 anafunctors. )
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 categorified subobject classifier: indeed, it is the subobject classifier in the 2-topos Cat.
For discussion of this point see
It was David Roberts who pointed out in
the relation between these higher classifiers and higher generalized universal bundles, motivated by the observations on principal universal 1- and 2-bundles in
Last revised on February 28, 2021 at 00:59:33. See the history of this page for a list of all contributions to it.