nLab pointed set

Definition

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 of pointed sets

Definition

Definition

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

$$\array{ && {\ast} \\ & ^{s_1}\swarrow && \searrow^{s_2} \\ S_1 &&\to&& S_2 } \,.$$

The category $\Set_{\ast}$ naturally comes with a forgetful functor $p : \Set_{\ast} \to \Set$ which forgets the tip of these triangles.

Properties

Proposition

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.

Pointed objects in the category of pointed sets

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 $\mathbb{2} \to A$.

Natural numbers object

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:

$$\array{ \mathbb{2} & \stackrel{z_0}{\to} & \overline{\mathbb{N}} & \stackrel{z_s}{\leftarrow} & \overline{\mathbb{N}} \\ & \mathllap{f} \searrow & \downarrow \mathrlap{\phi_{f, g}} & & \downarrow \mathrlap{\phi_{f, g}} \\ & & A & \underset{g}{\leftarrow} & A }$$

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

$$(-)+(-):\overline{\mathbb{N}} \times \overline{\mathbb{N}} \to \overline{\mathbb{N}} \wedge \overline{\mathbb{N}} \to \overline{\mathbb{N}}$$

by

$$z_0(p) + z_0(q) = z_0(p \vee q) \qquad z_s(m) + z_0(q) = z_s(m + z_0(q))$$

$$z_0(p) + z_s(n) = z_s(z_0(p) + n) \qquad z_s(m) + z_s(n) = z_s(z_s(m + n))$$

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 $Set_*$.

Interpretation as universal Set-bundle

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

$$\array{ \Set_{\ast} &\to& \pt \\ \downarrow && \downarrow^{\pt \mapsto {\ast}} \\ [I,\Set] &\stackrel{d_0}{\to}& \Set \\ \downarrow^{d_1} \\ \Set }$$

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

• Kathryn Hess, Bundle theory for categories .

(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. )

Interpretation as 2-subobject-classfier

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

• David Corfield: 101 things to do with a 2-classifier (blog)

It was David Roberts who pointed out in

• David Roberts, comment to: 101 things to do with a 2-classifier

the relation between these higher classifiers and higher generalized universal bundles, motivated by the observations on principal universal 1- and 2-bundles in

• David Roberts, Urs Schreiber, The inner automorphism 3-group of a strict 2-group, Journal of Homotopy and Related Structures, Vol. 3(2008), No. 1, pp. 193-244, (arXiv).

Related concepts

• pointed object

• pointed topological space

• pointed simplicial set

• pointed homotopy type