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 tensor unit of pointed sets is the boolean domain $\mathbb{2}$, and pointed objects in the category of pointed sets are pointed sets $A$ with a point-preserving function $\mathbb{2} \to A$.
In classical mathematics, the natural numbers object in $Set_*$ is the set of extended natural numbers $\overline{\mathbb{N}} = \mathbb{N} + \{\infty\}$, and comes with point-preserving functions $z_0:\mathbb{2} \to \overline{\mathbb{N}}$ and $z_s:\overline{\mathbb{N}} \to \overline{\mathbb{N}}$ such that for all pointed sets $A$ and point-preserving functions $f:\mathbb{2} \to A$, $g: A \to A$, there is a unique point-preserving function $\phi_{f, g}:\overline{\mathbb{N}} \to A$ making the following diagram commute:
The point-preserving function $z_0$ represents the function which takes the boolean true to $\infty$ and false to zero, and $z_s$ represents the point-preserving function which takes a natural number to its successor and $\infty$ to $\infty$.
The absorption monoid structure on $\overline{\mathbb{N}}$ is defined by double induction on $\overline{\mathbb{N}}$, we define
by
for all $p, q \in \mathbb{2}$ and $m, n \in \overline{\mathbb{N}}$, where $p \vee q$ is the disjunction of booleans $p$ and $q$ (recall the definition of addition in the natural numbers, inductively defined by $0(p) + 0(q) = 0(p \cdot q)$, $s(m) + 0(p) = s(m + 0(p))$, $0(p) + s(n) = s(0(p) + n)$, and $s(m) + s(n) = s(s(m + n))$ for all $p, q \in \mathbb{1}$ and $m, n \in \mathbb{N}$). It is a commutative monoid and represents addition in $\overline{\mathbb{N}}$.
In constructive mathematics, the extended natural numbers $\overline{\mathbb{N}}$ and the disjoint union $\mathbb{N} + \{\infty\}$ are no longer the same; it is $\mathbb{N} + \{\infty\}$ which remains the natural numbers object in $Set_*$.
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 July 25, 2023 at 10:52:16. See the history of this page for a list of all contributions to it.